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If x=a(theta+sintheta) , y=a(1+costheta)...

If `x=a(theta+sintheta)` , `y=a(1+costheta)` , prove that `(d^2y)/(dx^2)=-a/(y^2)` .

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Given, `x=a(theta+sintheta)` , `y=a(1+costheta)`

`(dx)/(d theta)=a(1+cos theta)`

`(dy)/(d theta)=a(-sin theta)`

`(dy)/(dx)=(-asin theta)/(a(1+cos theta))=(-2sin (theta/2) cos (theta/2))/(2cos^2 (theta/2))=-tan (theta/2)`

`(d^2y)/(dx^2)=((1+cos theta) cos theta -sin theta (0-sin theta))/(1+cos theta)^2 *(d theta)/(dx)`

`(d^2y)/(dx^2)=-(cos theta +cos ^2 theta + sin ^2 theta )/(1+cos theta)^2 *(1)/(a(1+cos theta))`

`(d^2y)/(dx^2)=-(cos theta +1)/(1+cos theta)^2 *(1)/(a(1+cos theta))`

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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
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  2. If x=a(1-cos^3theta) , y=a\ s in^3theta , prove that (d^2y)/(dx^2)=(32...

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  3. If x=a(theta+sintheta) , y=a(1+costheta) , prove that (d^2y)/(dx^2)=-a...

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  4. If x=a(theta-sintheta) , y=a(1+costheta) find (d^2y)/(dx^2)

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  5. If x=a(1-costheta) , y=a(theta+sintheta) , prove that (d^2y)/(dx^2)=-1...

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  6. If x=a(1+costheta) , y=a(theta+sintheta) , prove that (d^2y)/(dx^2)=(-...

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  7. If x=costheta , y=sin^3theta , prove that y\ (d^2y)/(dx^2)+((dy)/(dx))...

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  8. If y=sin(sinx) , prove that (d^2y)/(dx^2)+tanxdot(dy)/(dx)+y\ cos^2x=0...

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  9. If x=sint ,y=sinp t ,"p r o v et h a t" (1-x^2)(d^2y)/(dx^2)-x(dy)/...

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

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

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  12. If y=3cos(logx)+4sin(logx),\ prove that x^2y2+x\ y1+y=0 .

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

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  14. If x=sin(1/alogy) , show that (1-x^2)y2-x\ y1-a^2\ y=0 .

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  15. If logy=tan^(-1)x , show that (1+x^2)y2+(2x-1)y1=0 .

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  16. If y=tan^(-1)x , show that (1+x^2)(d^2y)/(dx^2)+2x(dy)/(dx)=0 .

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

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  18. If y=(tan^(-1)x)^2 , then prove that (1+x^2)^2\ y2+2x\ (1+x^2)y1=2 .

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  19. If y=cotx show that (d^2y)/(dx^2)+2y(dy)/(dx)=0 .

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  20. Find (d^2y)/(dx^2) , where y=log((x^2)/(e^2)) .

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