2nd ORDER DERIVATIVES

   2nd ORDER DIFFERENTION

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1) y=log(ax+b) prove
     y₂= -a²/(ax+b)²

2) log(x)/x prove x³y₂ - 2log(x)+3=0

3) y= xᵐeⁿˣ show that

y₂= {m(m-1)xᵐ⁻² +
                 2mnxᵐ⁻¹+n²xᵐ}eⁿˣ

4) x=(1-t)/(1+t) and y= 2t/(1+t) prove y₂=0

5) x=t+1/t and y= t- 1/t show that at t=2 , y₂ =-32/27

6) x²/a² + y²/b²=1 show
     y₂=-b⁴/(a²y³)

7) If ax²+2hxy+by²=1 prove that
      y₁=-(ax+hy)/(hx+by)
      y₂ = (h² -ab)/(hx+by)³

8) y(1-x)=x² show (1-x)y₂- 2y₁ =2

9) If log(√(x-2) + √(x+2))
     prove (x²-4)y₂+xy₁=0

10) log(x+√(x²+1)) show
       (x²-1)y₂+xy₁=0

11) y=√(x+1) +√(x-1) prove that
     (x²-1)y₂+xy₁=y/4

12) y= px +q/x² show
       x²y₂+2xy₁ =2y

13) u=v³log(1/v) show
       vu₂- 2u₁+3v²=0

14) if x=sin(θ), y=sin(pθ) prove that
      (1-x²)y₂ - xy₁+p²y=0

15) if x= a(θ+sin(θ)), y=a(1-cosθ) prove that 4ay₂=sec⁴(θ/2)

16) if cos⁻¹(y/b) = n log(x/n) prove
       x²y₂ + xy₁ + n²y =0

17) if y= tanx + sec x
       show (1-sinx)²y₂= cosx.