if y = e^(4x)sin3x show that y''^2-8y'+25y = 0

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`y = e^(4x)sin3x`

`y' = e^(4x)3cos3x+sin3x4e^4x`

`y' = e^(4x)(3cos3x+4sin3x)`

`y'' = e^(4x)(-9sin3x+12cos3x)+(3cos3x+4sin3x)xx4e^(4x)`

`y'' = e^(4x)(-9sin3x+12cos3x+12cos3x+16sin3x)`

`y'' = e^(4x)(24cos3x+716sin3x)`

`y'' = 8e^(4x)(3cos3x+4sin3x)-25e^(4x)sin3x`

`y'' = 8y'-25y`

`y''-8y'+25y = 0`

*So the answer is proved.*

`y''-8y'+25y = 0`

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