If `y=e^(4x) sin3x` , show that `(d^2y)/dx^2-8 dy/dx+25y=0`

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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`y=e^(4x) sin3x`

`dy/dx = e^(4x) (cos3x)3+(sin3x)e^(4x)xx4`

`dy/dx = 3e^(4x) cos3x+4y`

` (d^2y)/(dx^2)=3(e^(4x)3(-sin3x)+cos3x*4e^(4x))+4dy/dx`

`(d^2y)/(dx^2) = -9e^(4x)sin3x+4dy/dx-16y+4dy/dx`   [Using ans. of `dy/dx`]

`(d^2y)/(dx^2) = -9y+8dy/dx-16y`

`(d^2y)/(dx^2)-8dy/dx+25y = 0`

So the answer is obtained as required.


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