EXERCISE - A
y₂ as d²y/dx² and y₁ as dy/dx
Find d²y/dx²:::
1) 4x³ - 3x²+1. 24x - 6
2) 1/(x+a). 2/(x+a)³
3) xˣ. xˣ{(1+ logx)² + 1/x}.
4) x³ + tanx. 6x + 2sec²x tanx
5) sin(logx). -[sin(logx)+ cos(logx)]/x².
6) log(sinx). - cosec²x.
7) eˣ sin 5x. 2eˣ(5 cos 5x - 12 sin 5x).
8) e⁶ˣ cos 3x. 9e⁶ˣ(3 cos 3x - 4 sin 3x).
9) x³ logx. x(5+ 6 logx)
10) x cosx. - x cosx - 2 sinx.
11) log (logx). -(1+logx)/(x logx)².
12) Log(1-x). -1/(1-x)²
13) Log{x²/e²}. -2/x²
14) If x= 4t²+ 5, y= 6t² + 7t +3. -7/64t³
EXERCISE - B
Prove::
1) If y= √(3x+2). Then y y₂ + y₁²=0
2) If y= y= (x+ √(1+x²)ⁿ, Prove that (1+x²) y₂ + x y₁= m²y.
3) If x²/a² + y²/b²= 1, show y₂ = - b⁴/(a²y³).
4) If ax² + 2hxy + by²= 1, show y₁ = -(ax+hy)/(hx+by) and y₂ =(h²-ab)/(hx+ by)³.
5) If y(1-x)= x², show (1-x) y₂- 2 y₁= 2.
6) If y= √(x+1) + √(x-1), then show that (x² - 1)y₂+ x y₁= y/4.
7) If y= px + q/x², then show that x²y₂ + 2xy₁= 2y.
8) If y² = p(x) be a polinomial in x of degree ≥ 3, then obtain 2y³y₂ = y²p"(x) - 1/2 {p'(x)}².
9) y= xˣ, show y₂ - 1/y (y₁)² - y/x= 0.
10) If (x-a)²+ (y -b)²= c², show that [{1+(y₁)²}³⁾²]/y₂ is a constant independent of a and b.
11) If y= (ax+b)/(cx+d), show that
2 dy/dx. d³y/dx³ = 3(d²y/dx²)².
EXERCISE - C
PROVE:
1) If y= A cos nx + B sin nx, then y₂ + n²y= 0.
2) If y= 3 cos x + 2 sin x, then y₂ + y= 0.
3) Find A and B so that y= A sin 3x + B cos 3x, then d²y/dx²+ 4 dy/dx + 3y = 10 cos 3x.
4) If y= cotx, show y₂ + 2yy₁= 0.
5) If y= x + tanx, show that
A) cos 2x y₂ -2y + 2x= 0.
B) cos²x y₂ -2y + 2x= 0.
6) If y= Cosecx + cotx, show sinx y₂ = y².
7) If y= a cos(logx) + b sin(logx), show x²y₂ + x y₁+ y= 0.
8) If y= 3 cos(logx) + 4 sin(logx), show x²y₂ + x y₁+ y= 0.
9) If y= tanx + secx, show (1- sinx)² y₂ = cosx.
10) y= tanx show y₂ = 2yy₁
11) If x = tan(1/a log y), show (1+ x²)y₂ + (2x - a)y₁ = 0.
12) If x = sin(1/a log y), show (1- x²)y₂ - xy₁ - a²y = 0.
13) If y= sin(sinx), show y₂ + tanx y₁ + y cos²x = 0.
14) If y= sin(logx), show x² d²y/dx² + x dy/dx + y= 0.
EXERCISE-D
PROVE::
1) If y= x³ log(1/x), then xy₂- 2 y₁+ 3x²= 0.
2) If y= x³ logx , then d⁴y/dx⁴= 6/x.
3) If y= log(sinx), show d³y/dx³= 2 cosx cosec³x.
4) If y= log{x + √(x² + a²)}, show that (a² + x²) y₂ + xy₁= 0.
5) If y= {log{x + √(x² + 1)}², show that (1+ x²) y₂ + xy₁= 4
6) If y= log(ax+b), show y₂= - a²/(ax + b)².
7) If y= (logx)/x, show x³y₂ - 2logx +3= 0.
8) If y= log{√(x-2) + √(x+2)}, then show (x² -4)y₂+ x y₁= 0.
9) If y= log{x +√(x²+1)}, then show that (x² -1)y₂ + x y₁= 0.
10) If u= v³ log(1/v), show v d²u/dv² - 2 du/dv + 3v² = 0.
11) If y= x log{x/(a+bx)}, show x³y₂ = (x y₁ -y)².
12) If y= log(1+ cosx), show d³y/dx³ + d²y/dx². dy/dx = 0.
EXERCISE -E
PROVE
1) y= aeᵐˣ+ beᵐˣ, show y₂ + = m²y.
2) y= xᵐ eⁿˣ, show y₂ ={m(m-1)xᵐ⁻² + 2mnxᵐ⁻¹ + n²xᵐ} eⁿˣ.
3) If eʸ(x+1) = 1, show y₂ = (y₁)².
4) If y= e⁻ˣ cosx, show y₂= 2e⁻ˣ sinx
5) If y= eˣ cosx, show y₂= 2eˣ cos(x +π/2).
6) If y= eˣ (ax+b) show y₂- 4y₁ + 4y= 0.
7) If y= e⁻ᵏᵗ cos(pt +c), show y₂ +2k y₁ + ny²= 0. Where n² = p² + k².
8) If y= a eˣ + be⁻ˣ show y₂ - y₁ -2y= 0.
9) If y= eˣ(sinx + cosx) show y₂ - 2y₁ + 2y= 0.
10) If y= ₑ a cos⁻¹x show (1-x²) d²y/dx² - x dy/dx - a²y = 0.
11) If y= 509 e⁷ˣ + 600e⁻⁷ˣ show y₂ = 49y
12) If y= 3e²ˣ + 2e³ˣ show y₂ - 5 y₁ +
6y= 0.
EXERCISE - F
PROVE
1) If y= tan⁻¹x, show. (1+x²)y₂+2x y₁ = 0.
2) If y= (tan⁻¹x)², show. (1+x²)²y₂+ 2x(1+x²) y₁ = 2.
3) ₑ m cos⁻¹x, show. (1- x²)y₂ - x y₁ = m²y.
4) If y= (sin⁻¹x)², show. (1- x²)y₂ - x y₁ = 2.
5) If cos⁻¹(y/b)= n log (x/n) show. x²y₂ + x y₁ + n²y = 0.
6) If sin⁻¹x= y, show. y₂ = -x/√(1-x²)³
7) If y= sin⁻¹x show.(1-x²) y₂ - y₁=0
8) If log y= tan⁻¹x show.(1+ x²) y₂ +(2x -1) y₁=0.
EXERCISE - G
PROVE
1) If x= at², y= 2at. 1/(2at³)
2) If x= a(1- cos t), y= a(t+ sint) at t= π/2. -1/a
3) If x= sint, y= sin(pt), show (1-x²)y₂ - xy₁ + p²y = 0.
4) If x= a(t + sint), y= a(1- cost), then show 4ay₂ = sec⁴(t/2)
5) If y= a(t + sint), x= a(1- cost), then show y₂ = -1/a at t=π/2
6) If x= a(1 + cost), y= a(t+ sint), then show y₂ = -1/a at t=π/2.
7) If x= a(t + sint), y= a(1 + cost), then show y₂ = - a/y².
8) If x= (1-t)/(1+t) and y= 2t/(1+t), show that y₂ = 0.
9) If x= t + 1/t and y= t - 1/t, then show at t=2 is y₂ = -32/27
10) x= a cost + b sint and y= a sint - b cost, show y² y₂ - xy₁ + y = 0.
11) x= a sin t - b cost and y= a cost + b sin t, show y₂ = -(x²+y²)/y³.
12) If x= a cos³t, y= a sin³t, show y₂ = 1/3a sec⁴ t cosec t.
13) If x= a sect and y= b tant show d²y/dx² = - b⁴/(a²y³).
14) If x= a(cost + t sint), y= a(sint - t cost), show d²y/dx²= sec³t/at.
15) If x= a cost, y= b sint , show that d²y/dx² = - b⁴/(a²y³).
16) If x= a(1- cos³t), y= a sin³t, show d²y/dx²= 32/27a at t= π/6.
17) If x= cost, y= sin³t, show yd²y/dx² +(dy/dx)²= 3 sin²t(5cos²t -1).
18) If y= 2 cost - cos 2t, y= 2 sint - sin 2t, at t=π/2 show y₂ = -coty cosec²y.
MISCELLANEOUS QUESTIONS:
PROVE::
1) If 2x= y¹⁾ᵐ+ y⁻¹⁾ᵐ, show (x²-1)y₂ + x y₁= m²y.
2) If 2x= y¹⁾⁵ + y⁻¹⁾⁵, show (x²-1)y₂ + x y₁= 25y.
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