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If x=asint-bcost ,y=acost+bsint ,"p r o...

If `x=asint-bcost ,y=acost+bsint ,"p r o v et h a t"` `(d^2y)/(dx^2)=-(x^2+y^2)/(y^3)`

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Here, `x = asint - bcost` and `y = acost+bsint`
`:. dy/dx = (dy/dt)/(dx/dt) = (-asint+bcost)/(acost+bsint)`
Now, `(d^2y)/dx^2 = (d/dt(dy/dx))/(dx/dt) `
`=((-acost-bsint)(acost+bsint)-(-asint+bcost)(-asint+bcost))/((acost+bsint)^2(acost+bsint)`
`=(-(acost+bsint)^2 - (bcost-asint)^2)/((acost+bsint)^3)`
`=(-x^2-y^2)/(y^3)`
`:. (d^2y)/dx^2 = (-(x^2+y^2))/(y^3).`
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
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  4. If y=e^(2x) (ax+b), show that y2-4y1+4y=0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  20. 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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