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If y=3cos(logx)+4sin(logx),\ prove that...

If `y=3cos(logx)+4sin(logx),\ ` prove that `x^2y_2+x\ y_1+y=0` .

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To prove that \( x^2 y_2 + x y_1 + y = 0 \) for the function \( y = 3 \cos(\log x) + 4 \sin(\log x) \), we will first calculate the first and second derivatives of \( y \). ### Step 1: Calculate \( y_1 \) Given: \[ y = 3 \cos(\log x) + 4 \sin(\log x) \] ...
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. If y=(sin^(-1)x)^2 , prove that (1-x^2)y2-x y1-2=0 .

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

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

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

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

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

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

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

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

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

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

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

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

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  15. If y=e^acos^((-1)x) , -1lt=xlt=1 , show that (1-x^2) (d^2y)/(dx^2)-x(d...

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  16. If y=500\ e^(7x)+600\ e^(-7x) , show that (d^2y)/(dx^2)=49\ y .

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  17. If x=2cost-cos2t ,\ \ y=2sint-sin2t , find (d^2y)/(dx^2) at t=pi/2 .

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  18. If x=4z^2+5,\ \ y=6z^2+7z+3, find (d^2y)/(dx^2) .

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

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  20. If y=sin(logx) , prove that x^2(d^2y)/(dx^2)+x(dy)/(dx)+y=0 .

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