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

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

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Here,
` x=a(1+cos theta) { and } y=a(theta+sin theta)`
Differentiating w . r.t. `theta`, we get
` frac{d x}{d theta}=-a sin theta { and } frac{d y}{d theta}=a+a cos theta `
` therefore frac{d y}{d x}=frac{a+a cos theta}{-a sin theta}=frac{1+cos theta}{-sin theta}`
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
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  13. If y=tan^(-1)x , show that (1+x^2)(d^2y)/(dx^2)+2x(dy)/(dx)=0 .

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

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

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  18. If y=a e^(2x)+b e^(-x) , show that, (d^2y)/(dx^2)-(dy)/(dx)-2y=0 .

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  19. If y=e^x(sinx+cosx) prove that (d^2y)/(dx^2)-2(dy)/(dx)+2y=0 .

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  20. If y=cos^(-1)x , find (d^2y)/(dx^2) in terms of y alone.

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