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If y=Acos(logx)+\ Bsin(logx) , prove tha...

If `y=Acos(logx)+\ Bsin(logx)` , prove that `x^2\ (d^2y)/(dx^2)+x(dy)/(dx)+y=0` .

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Verified by Experts

We have given
`y=Acos(logx)+\ Bsin(logx)`
Then we have to proof
`x^2\ (d^2y)/(dx^2)+x(dy)/(dx)+y=0`
As,
`y=Acos(logx)+\ Bsin(logx)`
On differentiating w.r.t. x, we get
`(dy)/(dx)=a([-sin(log x)])/(x)-(bcos(log x))/(x)`
...
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. If y=Acosn x+Bsinn x , show that (d^2y)/(dx^2)+n^2\ y=0 .

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  2. If y=A e^(m x)+B e^(n x) , show that (d^2y)/(dx^2)-(m+n)(dy)/(dx)+m n ...

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  3. If y=Acos(logx)+\ Bsin(logx) , prove that x^2\ (d^2y)/(dx^2)+x(dy)/(dx...

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  4. If y=tanx+secx , prove that (d^2y)/(dx^2)=(cosx)/((1-sinx)^2)

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  5. If y=tanx , prove that y2=2y y1 .

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

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

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  8. If y^3-y=2x , prove that (d^2y)/(dx^2)=-(24 y)/((3y^2-1)^3) .

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  9. If e^y(x+1)=1 . Show that (d^2y)/(dx^2)=((dy)/(dx))^2

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  10. If y=x^x , find (d^2y)/(dx^2) .

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  11. If y=xlog[x/((a+b x))] , then show that x^3(d^2y)/(dx^2)=(x(dy)/(dx)-y...

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  12. If y=log{x+ sqrt( x^2+a^2 )} , prove that: (x^2+a^2)(d^2y)/(dx^2)+x(dy...

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  13. If y=sin^(-1)x , then show that (1-x^2)\ (d^2y)/(dx^2)-x(dy)/(dx)=0 .

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  14. If y=e^(m\ sin^(-1)x) , prove that (1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)-m^2...

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  15. If y={x+sqrt(x^2+1)}^m , show that (x^2+1)y2+x y1-m^2\ y=0

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  16. If y=(sin^(-1)x)/(sqrt(1-x^2)) , show that (1-x^2)(d^2y)/(dx^2)-3x(dy)...

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

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  18. If y=x^x , prove that (d^2y)/(dx^2)-1/y((dy)/(dx))^2-y/x=0

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  19. Find (d^2y)/(dx^2) , if x=a t^2,\ \ y=2a t .

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  20. If x=acos^3theta and y=asin^3theta, then find the value of (d^2y)/(...

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