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If x=costheta,ysin^3theta,"p r o v et h...

If `x=costheta,ysin^3theta,"p r o v et h a t"` `y(d^2y)/(dx^2)+((dy)/(dx))^2=3sin^2theta(5cos^2theta-1)dot`

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`dy/dx=(dy/dtheta)/(dx/dtheta)=(3sin^2thetacostheta)/(-sintheta)=-3sinthetacostheta`
`(d^2y)/(dx^2)=(-cos^2theta*3+3sin^2theta)/(-sinntheta)`
`y(d^2y)/(dx^2)=sin^3theta[3cos^2theta-3sin^2theta]+9sin^2thetacos^2theta`
`=3sin^2cos^2theta-3sin^4theta+9sin^2cos^2theta`
`=3sin^2theta[5cos^2theta-1]`.
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. Find (d^2y)/(dx^2)where y=log((x^2)/(e^2))dot

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  2. If x=costheta,ysin^3theta,"p r o v et h a t" y(d^2y)/(dx^2)+((dy)/(...

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

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  4. If x=a(1-cos^3theta),y=asin^3theta,"p r o v et h a t"(d^2y)/(dx^2)=(...

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  5. If x=a(theta+sintheta),y=a(1+costheta),y=a(1+costheta), Prove that (d...

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  6. If x=2cost-cos2t ,y=2sint-sin2t ,"f i n d"(d^2y)/(dx^2)"a t"t=pi/2dot

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  7. If y=log(1+cosx),prove that (d^3y)/(dx^3)+(d^2y)/(dx^2)dot(dy)/(dx)=0...

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  8. If y="sin"(sinx),"p r o v et h a t" (d^2y)/(dx^2)+tanx(dy)/(dx)cos^...

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  9. If x=sint and y=sinp t , prove that (1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)+p^...

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  10. If x=a(cost+tsint) and y=a(sint-tcost), then find the value of (d^2y)...

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  11. If y=(cot^(-1)x)^2,p rov et h a ty2(x^2+1)^2+2x(x^2+1)y1=2.

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  12. If y=e^xcosx ,p rov et h a t(d^2y)/(dx^2)=2e^xcos(x+pi/2)dot

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  13. If x=cos t + (log tant)/2, y=sin t, then find the value of (d^2y)/(dt^...

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  14. Find A and B so that y=Asin3x+bcos3x satisfies the equation (d^2y)/(d...

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

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

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

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

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

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

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