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If y=a e^(2x)+b e^(-x) , show that, (d^2...

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

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To show that \(\frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = 0\) for the function \(y = ae^{2x} + be^{-x}\), we will differentiate \(y\) twice and substitute the results into the equation. ### Step 1: Find the first derivative \(\frac{dy}{dx}\) Given: \[ y = ae^{2x} + be^{-x} \] ...
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RD SHARMA-HIGHER ORDER DERIVATIVES-Solved Examples And Exercises
  1. If y=cotx show that (d^2y)/(dx^2)+2y(dy)/(dx)=0 .

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

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

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

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

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

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

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

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

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

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

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  12. If y=3\ e^(2x)+2\ e^(3x) , prove that (d^2y)/(dx^2)-5(dy)/(dx)+6y=0 .

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  13. If y=(cot^(-1)x)^2 , prove that y2(x^2+1)^2+2x(x^2+1)y1=2 .

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  14. If y=cosec^(-1)x ,\ \ x >1 , then show that x(x^2-1)(d^2y)/(dx^2)+(2x^...

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  15. If x=cost+logtan(t/2),\ \ y=sint , then find the value of (d^2y)/(dt^2...

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  16. If x=asint and y=a(cost+logtant/2) , find (d^2y)/(dx^2) .

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

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  18. If x=a(cost+logtan(t/2)) , y=asint , evaluate (d^2y)/(dx^2) at t=pi/3 ...

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  19. If x=a(cos2t+2tsin2t) and y=a(sin2t-2tcos2t) , then find (d^2y)/(dx^2)...

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  20. If x=asint-bcost ,\ \ y=acost+bsint , prove that (d^2y)/(dx^2)=-(x^2+y...

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