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If x=sint and y=sinp t , prove that (1-x...

If `x=sint` and `y=sinp t ,` prove that `(1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^2y=0.`

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`dy/dx=(dy/dt)/(dx/dt)=(cosptp)/cost=(Pcospt)/cost`
`d^y/dx^2=(p^2(-sinpt)+sint*Pcospt)/(cos^2t*cost)`
`=(-P^2sinptcost)/cos^3t+(p sintcospt)/cos^3t`
`=(-p^2sinpt)/(1-x^2)+(ptantcospt)/(1-x^2)`
`(1-x^2)(d^2y)/(dx^2)-xdy/dx`
`=-p^2sinpt+ptantcospt-p sint/costcospt`
`=-p^2y`.
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
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  2. If y="sin"(sinx),"p r o v et h a t" (d^2y)/(dx^2)+tanx(dy)/(dx)cos^...

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  3. If x=sint and y=sinp t , prove that (1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^...

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  4. If x=a(cost+tsint) and y=a(sint-tcost), then find the value of (d^2y)...

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  5. If y=(cot^(-1)x)^2,p rov et h a ty2(x^2+1)^2+2x(x^2+1)y1=2.

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  6. If y=e^xcosx ,p rov et h a t(d^2y)/(dx^2)=2e^xcos(x+pi/2)dot

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  7. If x=cos t + (log tant)/2, y=sin t, then find the value of (d^2y)/(dt^...

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  8. Find A and B so that y=Asin3x+bcos3x satisfies the equation (d^2y)/(d...

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  9. If x=asint-bcost ,y=acost+bsint ,"p r o v et h a t" (d^2y)/(dx^2)=-...

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  10. If y=e^(2x) (ax+b), show that y2-4y1+4y=0

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  11. If x=asectheta,y=btantheta,"p r o v et h a t"(d^2y)/(dx^2)=-(b^4)/(a...

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

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

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

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  15. If x=sin(1/alogy),"s h o wt h a t"(1-x^2)y2-xy1-a^2y=0.

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

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  17. If y=e^-x cos x, show that (d^2y)/(dx^2)= 2e^-1 sin x

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

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  19. If y=x^3logx ,"p r o v et h a t"(d^4y)/(dx^4)=6/xdot

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  20. If y=(logx)/x ,s howt h a t(d^2y)/(dx^2)=(2logx-3)/(x^3)

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