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

If `y=e^"tan"^((-1)"x"),` prove that `(1+x^2)y_2+(2x-1)y_1=0.`

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To prove that \((1+x^2)y_2 + (2x-1)y_1 = 0\) for the function \(y = e^{\tan^{-1}(x)}\), we will first find the first and second derivatives of \(y\). ### Step 1: Find the first derivative \(y_1\) Given: \[ y = e^{\tan^{-1}(x)} \] Using the chain rule, we differentiate \(y\): ...
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. If y=e^(2x) (ax+b), show that y2-4y1+4y=0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  17. 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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  18. If y=x^x ,fin d(d^2y)/(dx^2)

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

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

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