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If x=sin theta,y=cosptheta,p rov et h a...

If `x=sin theta,y=cosptheta,p rov et h a t` `(1-x^2)y_2-x y_1+p^2y=0,w h e r ey_2=(d^2y)/(dx^2)a n dy_1=(dy)/(dx)`

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We have, `x=sinθ,y=cospθ`
`(dx)/(d theta)​=cosx=sqrt(1−x^2 )`
​`(dy)/(d theta)​=−p sin pθ=−psqrt(1−y^2)`
​ ∴`(dy)/(dx)​=y_1​=(-p(sqrt(1−y^2))/(sqrt(1−x^2))`
​​ ⇒`y_1^2​=p^2(1−y^2)`
...
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. If y=2 sin x+3cos x, show that (d^2y)/(dx^2)+y=0

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

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

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

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  6. 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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  7. If y=x^x ,fin d(d^2y)/(dx^2)

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

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

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

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  11. If y=(a x+b)/(x^2+c), then (2x y1+y) y3=

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  12. If y=(log)e(x/a)^x ,then x^3y2=

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  13. If sin^(-1) (1 – x) – 2sin^(-1) x = π / 2,then x is equal to

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  14. If y=2/(sqrt(a^2-b^2)){tan^(-1)((a-b)/(a+b))tanx/2} , then show that (...

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  15. If y=x^(n-1)lnx then x^2y2+(3-2n)x y1 is equal to -(n-1)^2y (b) (n...

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  16. If x y-(log)e y=1 satisfies the equation x(y y2+y1 2)-y2+lambday y1=0,...

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  17. If x=f(t )cost-f^(prime)(t)sinta n dy=f(t)sint+f^(prime)(t)cost ,t h ...

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  18. If x=f(t) and y=g(t), then (d^2y)/(dx^2) is equal to

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  19. Let f(x) be a polynomial.Then, the second order derivative of f(e^x) ...

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  20. If y=tan^-1((log(e/x^2))/(log(ex^2)))+tan^-1((3+2logx)/(1-6logx)), the...

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