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If y=(sin^(-1)x)^2 ,prove that (1-x^2...

If `y=(sin^(-1)x)^2` ,prove that `(1-x^2)y_2-x y_1-2=0.`

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To prove that \( (1-x^2)y_2 - xy_1 - 2 = 0 \) for \( y = (\sin^{-1} x)^2 \), we will follow these steps: ### Step 1: Find the first derivative \( y_1 \) Given: \[ y = (\sin^{-1} x)^2 \] ...
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RD SHARMA ENGLISH-HIGHER ORDER DERIVATIVES-All Questions
  1. If x=asectheta,y=btantheta,"prove that"(d^2y)/(dx^2)=-(b^4)/(a^2y^3)

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  2. If y=e^"tan"^((-1)"x")), prove that (1+x^2)y2+(2x-1)y1=0.

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  3. If y=(sin^(-1)x)^2 ,prove that (1-x^2)y2-x y1-2=0.

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  4. If y={"log"(x+sqrt(x^2+1))}^2,"show that"(1+x^2)(d^2y)/(dx^2)+x(dy)/(...

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  5. If x=sin(1/alogy),"show that"(1-x^2)y2-xy1-a^2y=0.

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  6. If y=x+tan x, show that cos^2 x (d^2y)/(dx^2)-2y+2x=0

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  7. If y=e^-x cos x, show that (d^2y)/(dx^2)= 2e^-x sin x

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  8. If y=log(sinx),"p r o v et h a t"(d^3)/(dx^3)=2cosxcos e c^3xdot

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  9. If y=x^3logx ,"prove that"(d^4y)/(dx^4)=6/xdot

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  10. If y=(logx)/x ,show\ tha t(d^2y)/(dx^2)=(2logx-3)/(x^3)

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  11. If y=2 sin x+3cos x, show that (d^2y)/(dx^2)+y=0

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  12. Find . (d^2y)/(dx^2),ifx=a t^2,y=2a t.

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  13. If x=sin theta,y=cosptheta,p rov e \ t h a t (1-x^2)y2-x y1+p^2y=0,w ...

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  14. If y=tan^-1x find (d^2y)/(dx^2) in terms of y alone.

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  15. If x^m y^n=(x+y)^(m+n), prove that (d^2y)/(dx^2)=0

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  16. If y^3-y=2x ,p rov et h a t(d^2y)/(dx^2)=(-24 y)/((3y^2-1)^3)

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  17. If y=x^x ,f i n d \ (d^2y)/(dx^2)

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  18. If y=sin^(-1)x , \ s how \ t h a t \ (d^2y)/(dx^2)=x/((1-x^2)^(3/2))

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  19. If y=Acosn x+Bsinn x ,s howt h a t (d^2y)/(dx^2)+n^2y=0

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  20. If y=tanx+secx ,p rov e \ t h a t \ (d^2y)/(dx^2)=(cosx)/((1-sinx)^2)

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