DIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATION: An equation containing an independent variable, dependent variable and differential coefficient of dependent variable with respect to independent variable is called a differential equation.
FOR INSTANCE,
1) dy/dx = 2xy
2) d²y/dx² = 4x
3) dy/dx = sinx + cosx.
4) dy/dx + 2xy = x³.
5) d²y/dx² - 5 dy/dx + 6y = x².
6) ³√{1 + (dy/dx)²}² = k d²y/dx²
7) y= x dy/dx + √{1+ (dy/dx)²}.
8) (x² + y²)dx - 2xy dy= 0.
9) (d³y/dx³) + (1+ dy/dx)³ = 0. are examples of differential equations.
ORDER OF DIFFERENTIAL EQUATIONS:
The order of differential equation is the order of the highest order derivative appearing in the equation.
EXAMPLE: d²y/dx² + 3 dy/dx + 2y eˣ,
the order of the highest order derivatives is 2. So, it is a differential equation of order 2. The equation d³y/dx³ - 6(dy/dx)² - 4y = 0 is of order 3, because the order of the highest order derivative in it is 3.
NOTE: The order of a differential equation is a positive integer.
DEGREE OF A DIFFERENTIAL EQUATION:
The degree of a differential equation is the degree of the highest order of derivative, when differential coefficients are made free from radicals and fractions.
In other words, the degree of a differential equation is the power of the highest order derivative occuring in a differential equation when it is written as a polynomial in differential coefficients.
EXAMPLE: d³y/dx³ - 6 (dy/dx)² - 4y = 0.
Here the power of highest order derivatives is 1. SO, it is a differential equation of degree 1.
EXAMPLE: x(d³y/dx³)²+ (dy/dx)⁴ + y² = 0.
Here the highest order derivative is 3 and its power is 2. So, it is differential equation of order 3 and degree 2.
EXAMPLE: y= x dy/dx + √{1+ (dy/dx)²}
When expressed as a polynomial in derivatives becomes (x² - 1)(dy/dx)² - 2xy dy/dx + (y² - 1) = 0.
Here the power of highest order derivative is 2. So, its degree is 2.
EXAMPLE: √{1+ (dy/dx)²}³= k(d²y/dx²)
the order of highest differential Coefficient is 2. So, its order is 2. To find its degree Express the differential equation as a polynomial in derivatives. When expressed as a polynomial in derivatives it becomes
k²(d²y/dx²)² - {1 + (dy/dx)²}³= 0.
Hete, the power of the highest differential Coefficient is 2. So, its degree is 2.
EXAMPLE: (x² + y²) dx - 2xy dy = 0.
Write as dy/dx = (x² + y²)/2xy.
So, it is a differential equation of order 1 and degree 1.
EXAMPLE: y= px + √(a²p² + b²), where p= dy/dx.
The order of the highest order derivative is 1. So, its order is 1. To determine its degree, express it as a polynomial in differential equation Coefficients as:
y= px + √(a²p² + b²)
=> (y - px)² = (a²p² + b²)
=> p²(x² - a²) - 2xyp + y² - b²= 0
=> (x² - a²) (dy/dx)² - 2xy(dy/dx) + y² - b²= 0.
So, the power of highest order differential Coefficient is 2. So, its degree is 2.
EXAMPLE: (d²y/dx²)² + sin(dy/dx)= 0.
The highest order derivative present in the differential equation is d²y/dx².
So, its order is 2. Since the differential equation cannot expressed as a polynomial in differential Coefficients. So, its degree is not defined.
Linear and nonlinear differential equations: If a differential equation when expressed in the form of a polynomial involves derivatives and dependent variable in the first power and there are no product of these, and also the coefficient of the various are either constant or function of the independent variable, then it is said to be linear differential equation. Otherwise, it is a non linear differential equation.
It follows from the above definition that a differential equation will be non-linear differential equation if
I) its degree is more than one.
II) Any of the differential Coefficient has exponents more than one.
III) exponents of the dependent variable is more than one.
IV) products containing dependent variable and its differential Coefficients are present.
EXAMPLE: The differential equation (d³y/dx³)³ - 6(d²y/dx²) - 4y = 0, is a non-linear differential equations, because it's degree is 3, more than one.
EXAMPLE: The differential equation d²y/dx² + 2(dy/dx)² + 9y = x, is a non-linear differential equation, because differential Coefficient dy/dx has exponent 2.
EXAMPLE: The differential equation (x² + y²)dx - 2xy dy = 0 is a non-linear differential equation, because the exponent of dependent variable y is 2 and it involves the product of y and dy/dx
EXAMPLE: Consider the differential equation d²y/dx² - 5 dy/dx + 6y = sinx. This is a linear differential equation of order 2 and degree 1.
EXAMPLE: [√{1+(dy/dx)²}³/d²y/dx² = K
Here,
K²(d²y/dx²)²= {1+ (dy/dx)²}³
The highest order differential coefficient in this equation is d²y/dx² and its power is 2. Therefore, the given differential equation is a non-linear differential equation of second order and second degree.
EXAMPLE: d²y/dx² = 1 + √(dy/dx)
Here,
(d²y/dx² -1)²= dy/dx
=> (d²y/dx²)² - 2 d²y/dx² - dy/dx + 1= 0
Clearly, it is non-linear differential equation of second order and second degree.
EXAMPLE: y= dy/dx + c/(dy/dx)
Here,
(dy/dx)² - y (dy/dx) + c = 0.
Clearly, it is a non-linear differential equation of order 1 and degree 2.
EXAMPLE: y + dy/dx = 1/4 ∫ y dx.
We have,
dy/dx + d²y/dx² = y/4 (diff. Both the sides)
Clearly, this is a differential equation of order 2 and degree 1. Also, it is a linear differential equation
EXAMPLE: dy/dx + sin(dy/dx)= 0.
Clearly, The highest order derivative present in the differential equation is dy/dx. So, it is of order 1.
Again LHS of the differential equation cannot be expressed as a polynomial in dy/dx.
So, its degree is not defined.
EXAMPLE: d⁵y/dx⁵ + ᵈʸ⁾ᵈˣ + y² = 0.
Clearly, The highest order differential coefficient present in the differential equation is d⁵y/dx⁵. So, it is of order 5.
Again LHS of the differential equation cannot be expressed as a polynomial in dy/dx.
So, its degree is not defined.
EXAMPLE: d⁴y/dx⁴ + sin(d³y/dx³)= 0
Clearly, The highest order derivative present in the differential equation is 4, so the order of the given differential equation is 4.
Again LHS of the differential equation cannot be expressed as a polynomial in dy/dx.
So, its degree is not defined.
EXAMPLE: (d²y/dx²)² + cos(dy/dx)= 0.
Clearly, The highest order derivative present in the differential equation is dy/dx. So, it is of order 2, so, its order is 2.
Again LHS of the differential equation cannot be expressed as a polynomial in dy/dx.
So, its degree is not defined.
EXERCISE--1(A)
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1) d³x/dt³ + d²x/dt² + (dx/dt)² = eᵗ
2) d²y/dx² + 4y = 0
3) (dy/dx)² + 1/(dy/dx) = 0
4) √{1 + (dy/dx)²} = ³√(c d²y/dx²).
5) d²y/dx² + (dy/dx)²+ xy = 0
6) ³√(d²y/dx²) = √(dy/dx).
7) d⁴y/dx⁴ = √{c+ (dy/dx)²}³
8) x + (dy/dx) = {1 + (dy/dx)²}.
9) y d²x/dy² = y² +1.
10) s² d²t/ds² + St dt/ds= s.
11) x² (d²y/dx²)³+ y(dy/dx)⁴+ y⁴= 0.
12) d³y/dx³ + (d²y/dx²)³+ dy/dx + 4y= sinx.
13) (xy² + x)dx + (y - x²y)dy = 0.
14) √(1- y²) dx + √(1- x²) dy = 0.
15) d²y/dx² = ³√(dy/dx)² +1.
16) 2 d²y/dx² + 3 √{1- (dy/dx)² - y}= 0.
17) 5 d²y/dx² = √{1+ (dy/dx)²}³.
18) y= x dy/dx + a √{1+ (dy/dx)²}.
19) y = px + √(a²p² + b²), where p= dy/dx.
20) dy/dx + eʸ = 0.
EXERCISE- 1(B)
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1) (d²y/dx²)² +(dy/dx)² = x sin(d²y/dx²)
2) (y")²+ (y')³ + siny = 0.
3) d²y/dx² + 5x (dy/dx) - 6y = log x.
4) d³y/dx³ + d²y/dx² + dy/dx + y siny = 0
5) d²y/dx² + 3 (dy/dx)²= x² log(d²y/dx²).
6) (dy/dx)³ - 4(dy/dx)²+ 7y = sinx.
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* FORMATION OF DIFFERENTIAL EQUATIONS:
STEP 1: Write the given equation involving independent variable x(say), dependent variable y (say) and the arbitrary constants.
STEP II: Obtain the number arbitrary constants in STEP I. Let there be n arbitrary constants.
STEP III: Differentiate the relation in STEP I n times with the respect to x.
STEP IV: Eliminate arbitrary constants with the help of n equation involving differential coefficient obt6in STEP III and equation in STEP I. The equation so obtained is the desired differential equation.
EXERCISE -2
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Type-1
* Form the differential equation of the family of curves represented :
1) y = eᵐˣ, where m is parameter. x dy/dx = y log y
2) y² = 4ax, where a is parameter. 2x dy/dx = y
3) y= mx + 5 where m is parameter. 2x dy/dx = y
4) xy = a², where a is parameter. y + x dy/dx = 0
5) x² + y² = a², a is parameter. x + y dy/dx = 0
6) x² - y² = a², a is parameter. x - y dy/dx = 0
7) x² + (y - b)² = 1, b are parameter. x + y dy/dx = 0
8) (2x + a)²+ y² = a², a is constant. y² - 4x² 2xy dy/dx = 0
9) (2x - a)²- y² = a², a is constant. 2x dy/dx = 4x² + y²
10) (x - a)²+ 2y² = a², a is constant. 2y² - x²= 4xy dy/dx
11) y² - 2ay + x² = a² by eliminating a. p²(x² - 2y²) - 4pxy - x² = 0, where p= dy/dx
12) y= a cos2x where a is a parameter. dy/dx + 2y tan2x = 0
13) y = c log x - 2 where c is a parameter. x dy/dx = y log y
Type-2
1) (x - a)² - y² = 1, a is parameter. y² (dy/dx)² - y² = 1
2) c(y + c)²= x³, where c is parameter. 8x (dy/dx)³ - 12y(dy/dx)² = 27x.
3) y= c(x - c)²= where c is parameter. (dy/dx)³= 4y(x dy/dx - 2y)
4) y² = (x - c)³, where c is parameter. 8(dy/dx)³= 27y
5) y = cx + 2c² + c³, where c is parameter. y= x dy/dx + 2(dy/dx)²+ (dy/dx)³
Type -3
1) Eliminate a,b,c from ax + by + c (b≠0). d²y/dx²=0
2) x/a + y/b = 1, where a and b are parameter. d²y/dx² =0
3) y = Aeˣ + Be⁻ˣ, where A, B are parameter. d²y/dx² - y = 0
4) y = Ae²ˣ + Be⁻²ˣ, where A, B are parameter. d²y/dx²= 4y
5) y = Aeˣ + Be⁻ˣ + x², where A, B are parameter. d²y/dx² - y = 2 - x²
6) y= A cos(x + B), where A, B are parameter. d²y/dx² + y = 0
7) Show x= a cos(kt + m) satisfies d²x/dt² + k²x = 0.
8) y= a sin(bx + c), where a and c are parameter. d²x/dt² + b²y=0
9) y= A sin mx + B cos mx where A and B are parameter . d²x/dx² + m²y=0
10) Eliminate A and B from A cos2x + B sin2x = e²ˣ/2. d²y/dx² + 4y= 4e²ˣ
11) x = A cos nt + B sin nt, where A, B are parameter. d²x/dt² + nx = 0
Type -4
1) y= ax + bx² where a, b are parameter. x² d²y/dx² - 2xy dy/dx + 2y= 0
2) y= ax + bx³ where a, b are parameter. x² d²y/dx² - 3xy dy/dx + 3y= 0
3) y= a tan⁻¹x + b where a, b are parameter. (1+x²)d²y/dx² + 2x dy/dx = 0
4) y= (ax +b)e⁻²ˣ where a, b are parameter.n. d²y/dx² + dy/dx + 4y= 0
5) x= e⁻ᵗ(a cost + b sin t) where a, b are parameter. d²x/dt² + 2dx/dt + 2x= 0
6) y= a secx + b tanx where a, b are parameter. d²y/dx² = dy/dx tanx + y sec²x
7) xy = Aeˣ + Be⁻ˣ, where A, B are parameter. x d²y/dx² + 2 dy/dx = xy
8) Find a differential equation which is satisfied by all the curves y=Ae²ˣ + Be⁻ˣ⁾², where A and B are non zero constants. 2 d²y/dx² - 3 dy/dx = 2y
Type -5
1) y²= a(b - x)(b + x), where a and b are parameter. x{y d²y/dx² + (dy/dx)²} = y dy/dx
2) y²= m(a² - x²), where a and m are parameter. x{y d²y/dx² + (dy/dx)²= y dy/dx
3) ax²+ by²=1 where a, b are parameter. x(y d²y/dx² + (dy/dx)²) = y dy/dx
Type -6
1) x²/a² + y²/b² = 1, where a and b are parameter. x(dy/dx)² + xy d²y/dx² - y dy/dx =0
2) 4(x - a) = (y- b)² = 1, where a and b are parameter. 2 d²y/dx² + (dy/dx)³ =0
3) y = ax²+ bx + c , where a, b, c are parameter. d³y/dx³ = 0
4) y² = a(b - x²), where a, b parameter. y dy/dx = x{y d²y/dx² + (dy/dx)²}
5) (x - a)² + (y - b)² = r², where a, b is parameter. (1+ p²)³= r²(d²y/dx²)², where p= dy/dx
6) (y - b)²= 4k(x - a) where a, b are parameter. 2k d²y/dx² + (dy/dx)³ = 0
Prove:
7) (x² - y²)= c (x² + y²)² is (x³ - 3xy²) dx= (y³ - 3x²y)dy where a and c are parameter.
Type -7
1) Find the differential equation of all circles touching the x-axis at the origin. (x²- y)² dy/dx = 2xy
2) Show that the differential equation dy/dx = y is found by eliminating a and b from the relation y= aeᵇ⁺ˣ. justify why the eliminate is of the first order although the relation involveds two constant a and b.
3) Prove that x = A cos√(πt) is a solution of the differential equation.
4) Show that the differential equation x(yd²y/dx² + (dy/dx)²) = y dy/dx is formed by eliminating a b and c from the relation ax² + by² + c =0. Justify why the eliminate is of the second order although the given relation involves three constants a, b and c.
5) Show that, the solution x = A cos(nt + B) + k/(n²+ p²). sin pt , for all A and B satisfies the differential equation d²y/dx² + n²x = k sin pt.
6) Show that the solution x= x= eᵏᵗ(a cos nt + b sin nt), for all a, b, always satisfies the differential equation d²x/dt² + 2k dx/dt + (k² + n²)x = 0.
7) Show that the solution y= a sinx + b cosx + x sinx satisfies. d²y/dx² + y = 2 cosx.
8) Show that, the equation of all circle touching the y-axis at the origin is, 2xy dy/dx = y²- x².
** Fill in the blanks:
1) if the number of independent arbitrary constants in the solution of a differential equation is equals to the order of the equation, then the solution is called ____.
b) The solution obtained by giving a particular value to the arbitrary constant of the complete solution is called a ____ of the differential equation.
c) The order and the degree of the differential equation (dy/dx)² - 2 dy/dx = 3x are ___ and ____respectively.
d) The order and the degree of the differential equation x dy/dx + x²y = 2 are ___ and ___ respectively.
e) The order and degree of the differential equation x² d²y/dx² - x dy/dx + y = 2x are ___ and ___respectively.
"" State whether the statement is true or false:
a) d²y/dx² - x (dy/dx)² + 2y = 0 is a differential equation of second order and first degree.
b) (d²y/dx²)² + (dy/dx)³ + y = 3x⁴ is a differential equation of degree three .
c) d³y/dx³+ y= √{1+dy/dx} is a differential equation of degree one.
d) The differential equation obtained by the elimination of arbitrary constant a and b from the equation y= a/x + b is of second order.
e) The differential equation obtained by eliminating a and b from the relation y= a. eᵇ⁺ˣ is of second order.
f) a third order differential equation is obtained by the elimination of arbitrary constants p, q and r from the equation px + qy+ r=0.
EXERCISE -3
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DIFFERENTIAL EQUATIONS OF THE TYPE dy/dx = f(x)
1) dy/dx = x/(x² +1). y= 1/2 log|x² + 1|
2) (x+ 2) dy/dx = x² + 4x - 9, x≠2. y= x²/2 + 2x - 13 log|x +2|
3) √(1- x⁴) dy = x dx. y= 1/2 sin⁻¹(x)²,
4) dy/dx = x² + x - 1/x, x≠ 0. y= x³/3 + x²/2 - log|x|
5) dy/dx = x⁵+x² -2/x, x≠ 0. y= x⁶/6 + x³/3 - 2 log|x|
6) (x² +1)dy/dx = 1. y= tan⁻¹x
7) (x +2)dy/dx = x² + 3x +7. y= x²/2 + x + 5 log|x +2|
8) √(a +x) dy + x dx = 0. y+ 2/3 √(a+ x)³ - 2a √(a+ x)
9) (x³ + x² + x +1) dy/dx = 2x² + x. y= 1/2 log|x +1| + 3/4 log(x² +1) - 1/2 tan⁻¹x
10) C'(x)= 2 + 0.15x; C(0)= 100. C(x)= 2x + (0.15) x²/2 + 100
11) x dy/dx +1= 0; y(-1)= 0. y= - log|x|
12) x(x²- 1) dy/dx = 1; y(2)= 0. y= log 4/3{(x²-1)/x²}
13) dy/dx= log x. y= x(log x -1)
14) dy/dx= x log x. y= x²/2 log x - x²/4
15) (eˣ+ e⁻ˣ) dy/dx= (eˣ- e⁻ˣ). y= log| (eˣ+ e⁻ˣ)|
16) dy/dx= (3e²ˣ+ 3e⁴ˣ)/(eˣ+ e⁻ˣ). y= e³ˣ
17) dy/dx + 2x = e³ˣ. y+ x²= 1/3 e³ˣ
18) dy/dx= x eˣ - 5/2 + cos²x. y= x eˣ - eˣ - 2x + 1/4 sin 2x
19) dy/dx= sin³x cos²x + xeˣ. Y= - 1/3 cos³x + 1/5 cos⁵x + x eˣ - eˣ
20) dy/dx = 1/(sin⁴x + cos⁴x). y= 1/√2 tan⁻¹{(tanx - cotx)/√2}.
21) dy/dx = (1- cosx)/(1+ cosx). y= 2 tan(x/2) - x + C
22) dy/dx = cos³x sin²x + x √(2x+1). y= 1/3 sin³x - 1/5 sin⁵x + 1/10 √(2x+1)⁵ - 1/6 √(2x+1)³
23) (sinx + cosx) dy + (cosx - sinx) dx = 0. y+ log|sinx + cosx| = C
24) dy/dx - x sin²x = 1/(x log x). y= x²/4 -(x sin 2x)/4 - 1/8 cos 2x + log|log x|
25) sin⁴x dy/dx = cos x. y= -1/3 cosec³x
26) Cosx dy/dx - cos 2x= cos 3x. y= sin 2x - x + 2 sinx - log|secx + tanx|
27) sin(dy/dx) = k; y(0)= 1. y- 1 = x sin⁻¹k
28) dy/dx = tan⁻¹x. y= x tan⁻¹x - 1/2 log|1+ x²|
29) 1/x dy/dx = tan⁻¹x, x≠0. y= 1/2 (x² +1) tan⁻¹x - x/2
30) dy/dx = x⁵ tan⁻¹(x³). y= 1/6(x⁶ tan⁻¹x³ - x³ + tan⁻¹x³)
31) (1 +x²) dy/dx - x = 2tan⁻¹x. y= 1/2 log|1+ x²| + (tan⁻¹x)²
32) ₑ(dy/dx) = x+1, y(0)= 5. y= x log(x +1) - x + log(x+ 1) +5
33) ₑ(dy/dx) = x+1, y(0)= 3. y= (x +1) log(x +1) - x +3
EXERCISE -4
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DIFFERENTIAL EQUATIONS OF THE TYPE dy/dx = f(y)
1) dy/dx + y = 1. x = - log|1- y|
2) dy/dx + (1+ y²)/y = 0. x + 1/2 log|1+ y²| = C
3) dy/dx = (1+ y²)/y³. x = y²/2 - 1/2 log|y² + 1|
4) dy/dx + 2y² = 0, y(1)= 1. y= 1/(2x -1)
5) dy/dx = 1/(y² + siny). x= y³/3 - cos y
6) dy/dx= sec y. x = siny
7) dy/dx = sin²y. x + cot y = C
8) dy/dx = (1- cos 2y)/(1+ cos 2y). x + cot y + y = C
EXERCISE - 5
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EQUATION IN VARIABLE SEPARABLE FORM:
Type-1
1) dx - dy + y dx + x dy =0. (x-1)²(y +1)²= c²
2) (x +1) dy/dx = 2xy. logx = 2{x - log|x +1|}+ C
3) (x -1) dy/dx = 2x³y. y = C|x -1|² ₑ(2x³/3 + x² + 2x)
4) (x - 1) dy/dx = 2x³y. Log|y| = 2x³/3 + x² + 2x + 2 log|x - 1| + C
5) (x -1) dy/dx = 2xy. logy = 2{x + log|x -1|}+ C
6) dy/dx = 1+ x + y+ xy. log|1+ y| = x + x²/2+ C
7) dy/dx = 1- x + y- xy. log|1+ y| = x - x²/2+ C
8) xy dy/dx = 1+ x + y+ xy. x + logx (1+ y)+ C = y
9) xy dy = (y -1)(x +1)dx. y - x = log|x| - log|y - 1|
10) dy + (x +1)(y +1)dx = 0. log|y +1| + x²/2 + x = C
11) x(x dy - y dx)= y dx, y(1)= 1. xₑ(1 - 1/x) ,x≠ 0.
12) x dy + y dx = xy dx,y(1)=1. y= eˣ ⁻¹
13) 2x dy/dx = 3y, y(1)=2. y² = 4x³
14) xy dy/dx = y +2, y(2)= 0. y - 2 log(y + 2) = log(x/8)
15) (xy² + 2x) dx + (x²y + 2y) dy = 0. y² + 2 = A/(x² + 2)
16) dy /dx =(x² +1)(y² +1). tan⁻¹y= x + x³/3 + C
17) (1+ x) (1+ y²) dx + (1+ y)(1+ x²)dy = 0. tan⁻¹x + tan⁻¹y + 1/2 log{(1+ x²)(1+ y²)}= C
18) (1+ x²) dy = xy dx. y= C √(1+ x²)
19) (1- x²) dy + xy dx = xy². log|y - 1)| - log|y|= - 1/2 log|1- x²|+ C
20) xy(y +1) dy = (x² +1) dx. y³/3 + y²/2 = x²/2 + log|x|
21) y - x dy/dx = a(y² + dy/dx). (x +a)(1- ay)= Cy
22) (y + xy) dx + (x - xy²)dy = 0. log x + x + log y - y²/2 = C
23) x dy/dx + y = y². y - 1 = C xy
24) x dy = (2x²+1)dx , y(1)=1. y³/3 = x² +5
25) y - x dy/dx = 2(1+ x² dy/dx), y(1)= 1. y= 2 - 3x)(2x +1)
26) y(1- x²) dy/dx = x(1+ y²). (1+ y²)(1- x²)= C.
27) (x² - y x²)dy + (y² + x²y²)dx = 0. log|y| + 1/y = -1/x + x + C
28) x √(1+ y²) dy/dx+ y √(1+ x²) = 0. √(1+ x²) + √(1+ y²) + 1/2 log|{√(1+ x²) - 1}/{√(1+ x²) + 1}| + 1/2 log |{√(1+ y²) -1}/{√(1+ y²)+1}|= C
29) dy/dx + √{(1- y²)/(1- x²)} = 0. y √(1- x²) + x √(1- y²)= C
30) √(1+ y² +x²+ x²y²) + xy dy/dx = 0. √(1+ x²) + √(1+ y²) + 1/2 log|{√(1+ x²) -1}/{√(1+ x²) +1}|= C
31) dy/dx = (1+ y²)/(1+ x²). y - x = C(1+ xy)
32) (y² +1) dx - (x² +1)dy = 0. tan⁻¹ x - tan⁻¹y = C
33) x (1+ y²) dx - y(1+ x²)dy = 0, y(1)= 0. (1+ x²)= 2(1+ y²)
Type- 2
1) dy/dx = (eˣ +1)y. log|y| = eˣ+ x
2) 5 dy/dx = eˣ y⁴. -5/(3y³) = eˣ +C
3) eˣ √(1- y²) dx + y/x dy = 0. xeˣ - eˣ = √(1- y²)+ C
4) dy/dx = eˣ⁻ʸ + x² e⁻ʸ. eʸ = eˣ + x³/3 + C
5) dy/dx = eˣ⁺ʸ. - e⁻ʸ = eˣ + C
6) dy/dx = eˣ⁺ʸ+ x²eʸ. - e⁻ʸ = eˣ + x³/3 +C
6) dy/dx = eˣ⁺ʸ+ x³eʸ. - e⁻ʸ +eˣ + x⁴/4 =C
7) y(1+ eˣ) dy = (y +1) eˣ dx. y - log|y + 1| = log|1+ eˣ| + C
8) dy/dx = eˣ⁺ʸ + e⁻ˣ⁺ʸ. e⁻ˣ - e⁻ʸ = eˣ + C
9) (1+ e²ˣ)dy + (1- y²)eˣ dx = 0 given when x= 0, y= 1. y= 1/eˣ
10) dy/dx = e²ˣ⁺ʸ dx, y(0)= 0. y= log{2/(3 - e²ˣ)
11) (x +1) dy/dx = 2e⁻ʸ - 1, y(0)= 0. y= log (2 - 1/(x +1)
12) dy/dx = 2eˣ y³ +2, y(0)= 1/2. y² (8- 4eˣ = 1
13) log(dy/dx)= ax + by. - 1/b e⁻ᵇʸ = 1/a eᵃˣ + C
14) (1+ y²)(1+ log x)dx + x dy = 0, y(1)= 1. y= tan{π/4 + 1/2 - 1/2 (1+ logx)}
15) log(dy/dx)= 3x + 4y, y(0)= 0. 4 e³ˣ + 3 e⁻⁴ʸ -7= 0
Type -3
1) cosx(1+ cosy)dx - siny (1+ sin x) dy = 0. (1+ sinx)(1+ cosy)= C
2) Cosx cos y dy/dx = siny sinx. siny = C cosx
3) dy/dx + (cosx siny)/cosy = 0. log|sin y| = - sinx + C
4) x cos²y dx = y cos²x dy. x tanx - y tany = log|secx| - log|secy|
5) x dy/dx + coty = 0. x= C cos y
6) tan y dx + sec²y tanx dy = 0. sinx tany = C
7) tan y dy/dx = sin(x+y)+ sin(x - y). 2 cosx secy = C
8) sec²x tan y dx + sec²y tanx dy = 0. |tanx tan y|= C
9) dy/dx = {x(2 logx +1)}/(siny + y cos y). y siny = x² logx + C
10) sin³x dy/dx = siny. cosy - 3/4 cosx + 1/12 cos 3x = C
11) cosecx log y dy/dx + x²y² = 0. - {(1+ logy)/y} - x² cosx + 2(x sinx + cosx)+ C
12) dy/dx = (cos²x - sin²x)cos²y. tan y = 1/2 sin 2x + C
13) dy/dx = y cot 2x, y(π/4)= 2. y² = 4 sin 2x
14) dy/dx = y sin 2x, y(0)=1. y=ₑsin²x
15) Sinx cosy dx + cosx siny dy = 0. y(0)=π/4. y= cos⁻¹ (1/√2 secx)
16) dy/dx= y tan 2x, y(0)=2. y= 2/√cos2x
17) dy/dx = y sin 2x, y(0)= 1. y = ₑsin²x
18) dy/dx = y tanx, y(0)= 1.
19) 3 eˣ tan y dx + (1- eˣ) sec²y dy = 0. (eˣ - 1)³= C tan x
20) x cosy dy = (xeˣ logx + eˣ) dx. Siny = eˣ logx
21) dy/dx = (xeˣ logx + eˣ)/(x cosy). Siny = eˣ logx
22) dy/dx ={eˣ(sin²x + sin2x)}/{y(2 logy +1)}. y² logy = eˣ sin²x + C
23) (eʸ +1) cosx dx + eʸ sinx dy= 0. (eʸ +1) sinx
EXERCISE - 6
EQUATION REDUCIBLE TO VARIABLE SEPARABLE FORM:
1) dy/dx= (4x +y +1)². 1/2 tan⁻¹{(4x+y+1)/2}= x + C
2) (x + y)² dy/dx= a². (x+ y) - a tan⁻¹{(x+y)/a} = x + C
3) (x + y +1)²= dy/dx. tan⁻¹{(x+y +1)} = x + C
4) dy/dx ={(x - y) +3}/{2(x - y)+5}. 2(x- y) + log(x - y +2) = x+ C
5) (x + y)²= dy/dx. (x + y)= tan{(x+ C)}
6) (x + y)²dy/dx = 1. y =tan⁻{(x+ y)}
7) (x+ y)(dx - dy)= dx + dy. 1/2 (y - x) + 1/2 log|x + y| = C
8) (x + y +1) dy/dx = 1. x= C eʸ - y - 2
9) dy/dx = eˣ⁺ʸ. - e⁻(ˣ⁺ʸ⁾ = x + C
10) dy/dx= Cos(x +y). tan{(x+y)/2} = x + C
11) dy/dx Cos(x - y) = 1. cot{(x - y)/2} = y + C
12) dy/dx= Cos(x +y) + sin(x + y). log|1+ tan{(x+y)/2}| = x + C
13) Cos²(x + 2y) - 2 dy/dx. tan(2x - 2y) + C= x
14) dy/dx = sec(x +y). y= tan{(x+y)/2}+ C
15) dy/dx = tan(x +y). y - x + log|sin(x + y)+ cos(x + y)|= C
16) sin⁻¹(dy/dx)= x+ y. x= tan(x +y) - sec(x +y)+ C
17) Cos(x +y)dy = dx, y(0)=0. y= tan{(x+y)/2}
18) (x +y +1)²dy = dx, y(-1)= 0. tan y= (x+y+1)
19) (x - y)(dy +dx)= dx - dy, y(0)= -1. x - y = eˣ⁺ʸ⁺¹
EXERCISE - 7
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HOMOGENEOUS DIFFERENTIAL EQUATIONS:
1) x²y dx - (x³ + y³) dy = 0. - x³/3y³ + log |y| = C
2) (x³ - 3xy²)dy = (y³ - 3x²y) dy. x² - y² = (x² + y²)² C²
3) x dy - y dx = √(x² + y²) dx. {y + √(x² + y²)}² = C²x⁴
4) x² dy + y(x + y) dx = 0. x²y = C(y + 2x)
5) dy/dx = (y - x)/(y + x). Log(x² + y²) + 2 tan⁻¹(y/x)= k
6) dy/dx = (y² - x²)/2xy. x² + y²= Cx
7) x dy/dx = y + x. y= x Log|x| + Cx
8) (x² - y²)dx - 2xy dy=0. x(x² - 3y²) = k
9) dy/dx = (y + x)/(x - y). tan⁻¹(y/x)= 1/2 log (x² + y²)+ C
10) 2xy dy/dx = x²+ y². C= 3x²y + 2y³
11) x² dy/dx = x² - 2y² + xy. (x +√2 y)/(x - √2 y) = (Cx²)^√2
12) xy dy/dx = x² - y². x²(x² - 2y²)= C
13) x² dy/dx = x² + xy + y². tan⁻¹(y/x) = C + log|x|
14) (y² - 2xy)dx= (x² - 2xy) dy. x²y - xy² = k
15) 2xy dx + (x² + 2y²)dy= 0. 3x²y + 2y³= C
16) 3x² dy = (3xy+ y²) dx. - 3x/y= log|x| + C
17) dy/dx = x/(2y+x). (x +y)(2y - x)²= C
18) (x + 2y)dx - (2x - y)dy= 0. C ₑ 2 tan⁻¹(y/x) = √(x² +y²)
19) dy/dx = y/x - √(y²/x² -1). y + √(y² - x²) = C
20) y² dx + (x² - xy + y²)dy = 0. y= C ₑ tan⁻¹(y/x)
21) [x √(x² + y²) - y²]dx + xy dy = 0. √(x² + y²) = x log|C/x|
22) (x² + y²) dy/dx = 8x² - 3xy + 2y². C ⁸√|(2x - y)|⁵ ¹⁶√(4x² + y²)³ = ₑ -3/8 tan⁻¹(y/2x)
23) (x² - 2xy) dy + (x² - 3xy + 2y²)dx = 0. y/x + logx = C
24) x dy/dx - y = 2 √(y² - x²). y + √(y² - x²) = C x³
25) (x² + 3xy + y²) dx - x² dy = 0. x/(x + y) + log x = C
26) (x - y) dy/dx = x + 2y. Log|x² + xy + y²| = 2√3 tan⁻¹{(x +2y/√3 x)}
27) (2x²y + y³) dx + (xy² - 3x³)dy = 0. x² y¹² = C⁴|2y²- x²|⁵
28) y dx + {x log(y/x)}dy - 2x dy = 0. Cy sin(x/y) = log|y/x| - 1
29) dy/dx = y/x {log y - logx +1}. log(y/x)= C
30) yx log(x/y)dx + {y² - x² log(x/y)}dy= 0. x²/y² {log(x/y)} + log y² = C
31) (1+ eˣ⁾ʸ) dx + eˣ⁾ʸ(1- x/y) dy = 0. x + y eˣ⁾ʸ= C
32) 2y eˣ⁾ʸ dx + (y - 2x eˣ⁾ʸ)dy = 0. 2 eˣ⁾ʸ = log|C/y|
33) y eˣ⁾ʸ dx = (x eˣ⁾ʸ + y)dy. eˣ⁾ʸ = log y + C
34) y{x cos(y/x) + y sin(y/x)} dx - x{y sin(y/x) - x cos(y/x)} dy = 0. | xy cos(y/x)|= k
35) x dy/dx = y - x tan(y/x). |sin(y/x)| = |C/x|
36)dy/dx = y/x+ sin(y/x). Tan(y/2x)= Cx
37) x dy/dx = y - x cos²(y/x). tan(y/x)= log|C/x|
38) y/x cos(y/x)dx - {x/y sin(y/x) + cos(y/x)} dy= 0. |y sin(y/x)| = C
39) x dy/dx = y - x cos²(y/x). tan(y/x) = log|C/x|
40) x cos(y/x) (y dx + x dy) = y sin(y/x)(x dy - dx) = 0. sec(y/x)| = C xy
41) x dy/dx - y +x sin(y/x) = 0. x sin(x/y) = C(1 + cos(y/x)
42) (x + y) dy + (x - y) dx = 0, y=1, x= 1. log(x² +y²) + 2 tan⁻¹(y/x) =π/2 + log 2
43) x² dy + y(x + y) dx = 0, y=1, x= 1. y = 2x/(3x² - 1)
44) (x² - y²) dx + 2xy dy = 0, y=1, x= 1. (x² + y²) = C |x|
45) (x² + xy)dy = (x² + y²) dx, y=0, x= 1. (x - y)² = |x| e⁻ʸ⁾ˣ
46) (3xy + y²)dx + (x² + xy) dy = 0, y=1, x= 1. |y² - 2xy| = C/x²
47) 2x² dy/dx - 2yx+ y² = 0, y(e)= e. y= 2x{1 - log|x|}
48) (x²+ y²) dx+ xy dy = 0, y(1)= 1. (2y³ + x²)x²= |C|
49) (x²- 2y²) dx+ 2xy dy = 0, y(1)= 1. y²= - x² log|x| + x²
50) (x²+ y²) dx = 2xy dy, y(1)= 0. (x² - y²)= x
51) (xy - y²)dx - x² dy = 0, y(1)= 1. y= x/{1+ log|x|}
52) dy/dx = {y(x + 2y)}/{x(2x + y)}, y(1)= 2. xy = 2 ³√|y - x|³
53) (y⁴ - 2x³y) dx + (x⁴ - 2xy³) dy = 0, y(1)=1. x³ + y³ = 2xy
54) x(3y² +x²) dx + y(3x² + y³) dy = 0, y(1)=1. x⁴ + 6x²y²+ y⁴ =8
55) (x - y) dy/dx = x + 2y x= 1, y= 0. Log{(x² + xy + y²)/x}= 2 √3 tan⁻¹{(x + 2y)/√3 x} - π/√3
56) dy/dx = xy/(x² + y²), y= 1, x =0. Log y = x²/2y²
57) (xeʸ⁾ˣ + y) dx = x dy, y(1)= 1. y= x - x log(1- e log|x|)
58) xeʸ⁾ˣ - y) + x dy/dx = 0, y(e)= 0. y= x log(log|x|)
59) x dy/dx sin(y/x)+ x - y sin(y/x)= 0, y(1)= π/2. e⁻ʸ⁾ˣ{sin(y/x) + cos (y/x)}= log|x|² + 2
60) dy/dx - y/x + cosec(y/x) = 0, y(1)= 0. log|x| = cos(y/x) - 1
61) {x sin²(y/x) - y}dx + x dy = 0, y(1) = π/4
62) x dy/dx - y + x sin(y/x) = 0, y(2)= x. cot(y/x) = log(ex)
63) x cos(y/x) dy/dx = y cos(y/x), x= 1, y = π/4. Sin(y/x) = log x + 1/√2
EXERCISE-8
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Type -1
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1) dy/dx - y/x = 2x². y= x³ + Cx
2) dy/dx + y/(2x) = 3x². y= 6x³/7 + C/√x
3) x dy/dx = x + y. y/x = log|x| + C
4) dy/dx +2 y= 4x. y= (2x- 1) + C e⁻²ˣ
5) dy/dx + 4xy/(x² +1)+ 1/(x² +1)²= 0. y(x²+ 1)²= - x + C
6) dy/dx + y/x = x³. 5xy = x⁵ + C
7) y² dx/dy + x - 1/y = 0. y= {(y+1)/y} + Cₑ1/y
8) (2x - 10y³) dy/dx + y= 0. x= 2y³ + C/y²
9) (x² -1)dy/dx + 2yx = 1/(x²- 1). y(x² -1)= 1/2 log|x-1)/(x+ 1)| + C
10) (x² +1) dy/dx + 2yx = √(x²+4). y(x²+1) = x/2 √(x² +4) + 2 logx|x + √(x² +4)|+ C
11) (1+ x²) dy/dx + 2yx - 4x² =0, y(0)= 0. y= 4x³/{3(x² +1)}
12) y dx - (x + 2y²) dy =0. x/y = 2y + C
13) y dx + (x + y³) dy =0. x= y³ + Cy
14) (1+ x²) dy/dx - 2xy = (x² + 2)(x² +1). y= (x + tan⁻²x + C)(x² +1)
15) (x² - 1) dy/dx + 2(x + 2)y = 2(x + 1). y= 2(x + 1)/(x - 1)³ {x² - 6x + 8 log(x +1)}+ C
Type -2
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1) x log x dy/dx + y = 2/x logx. y logx = - 2/x(1+ logx)+ C
2) x dy/dx + y = x log x. 4xy = 2x² log|x| - x² + C
3) dy/dx + 2y = e³ˣ. y= 1/5 e³ˣ + C e⁻²ˣ
4) 4 dy/dx + 8y = 5 e⁻³ˣ. y= -5/4 e⁻³ˣ + C e⁻²ˣ
5) dy/dx + 2y = 6eˣ. y e²ˣ= 2e³ˣ + C
6) dy/dx + y = e⁻²ˣ. y= eᵅˣ/(m +3) + C e⁻²ˣ
7) dy/dx + 3y = eᵅˣ. y= - e⁻²ˣ + C e⁻³ˣ
8) x dy/dx + y= xeˣ. y= {(x- 1)/x} eˣ+ C/x
9) x dy/dx - y = (x -1)eˣ. y= eˣ+ Cx
10) dy/dx - y = x eˣ. y= (x²/2 + C)eˣ
11) dy/dx + 2y = x e⁴ˣ. y= x/6 e⁴ˣ- 1/36 e⁴ˣ + C e⁻²ˣ
12) {ₑ- 2√x/√x - y/√x} dx/dy = 1. y= (2√x + C) ₑ- 2√x
Type -3
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1) dy/dx + y = sinx. y= Ce⁻ˣ + 1/2 (sinx - cosx)
2) dy/dx + y = cos x. y= Ce⁻ˣ + 1/2 (cosx - sinx)
3) dy/dx + 2y = sinx. y= Ce⁻²ˣ + 1/5 (2sinx - cosx)
4) dy/dx = y tanx - 2sinx. 2y cosx = cos 2x
5) dy/dx + y cosx = 2sinxcosx. y = sin x - 1 + C e⁻ˢᶦⁿˣ
6) x dy/dx + 2 y = x cosx. x²y = x² sin x + 2x cosx - 2 sinx + C
7) (1+ x²) dy/dx + y = tan⁻¹x. dy/dx + y = sinx. y= tan⁻¹ - 1+ Cₑ (- tan⁻¹x)
8) dy/dx + y tanx = cos x. y secx = x + C
9) dy/dx + y cotx = x² cotx + 2x. y sinx = x² sinx + C
10) dy/dx + y tan x = x² co²x. y secx = x² sinx + 2x cosx - 2 sinx +C
11) (1+ x²) dy/dx + y = ₑ ( tan⁻¹x). 2x ₑ (tan⁻¹x) = ₑ (2tan⁻¹y) + C
12) (Sinx) dy/dx + y cosx = 2 sin²x cosx. y sin x = 2/3 sin³x + C
Type -4
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1) dy/dx + y sec x = tanx. y(sec x + tanx)= secx + tanx - x + C
2) Cos²x dy/dx + y = tanx. y eᵗᵃⁿˣ= eᵗᵃⁿˣ(tanx - 1)+ C
3) dy/dx - 2y = cos 3x. ye⁻²ˣ = e⁻²ˣ/13 (3 sin3x - 2 cos 3x)+ C
4) dx + x dy= e⁻ʸ sec²y dy. xeʸ= tany + C
5) dy/dx +y = cosx - sinx. yeˣ = eˣ cos x + C
6) dy/dx + y tanx = 2x + x² tanx. y secx = x² sec x + C
7) dy/dx + y/x = cosx + 1/x sinx. y = sinx + C/x
8) dy/dx + x sin 2y = x³ cos²y. ₑx² tany = 1/2 (x² -1) ₑx² + C
9) dy/dx = -(x + y cosx)/(1+ sinx). y = (2C -x²)/{2(1+ sinx)}
10) x dy/dx + y - xy cotx = 0. xy sinx = - x cosx + sinx + C
11) (x + tany) dy = sin 2y dx. x= tany + C √tany
12) (1+ x²) dy + 2xy dx = cotx. y(x² +1) = log|sinx| + C
13) y+ d/dx (xy)= x(sinx + logx). y x² = - x² cosx + 2(x sinx + cosx) + x³/3 log x - x³/9+ C
14) dy/dx - y = eˣ, y(0)= 1. y= (x +1)eˣ
15) x dy/dx + y = x log x, y(1)= 1/4. y= x/2 logx - x/4 + 1/2x
16) dy/dx + 2x/(x² +1) = 1/(x² +1)², y(0)= 0. y= tan⁻¹x/(x² +1)
17) (x² +1) dy/dx - 2xy= (x⁴+ 2x² +1) cosx, y(0)= 0. y = (x² +1) sinx
18) (x - siny) dy + (tan y) dx = 0, y(0)= 0. y= sin⁻¹2x
19) (1+ y²) dx = (tan⁻¹y - x)dy, y(0)= 0. (x - tan⁻¹y +1) ₑ tan⁻¹y = 1
20) y eʸ dx = (y³ + 2x eʸ) dy, y(0)= 1. x= y²(1/e - 1/eʸ)
21) √(1- y²) dx = (sin⁻¹y - x) dy, y(0)= 0. x - sin⁻¹y + ₑ - sin⁻¹y
22) (x + 2y²) dy/dx = y, y(2)= 1. x= 2y²
