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If y = e^ax. cos bx, then prove that (d^2y)/(dx^2)-2ady/dx+(a^2+b^2)y=0 - Mathematics and Statistics

If y = eax. cos bx, then prove that

`(d^2y)/(dx^2)-2ady/dx+(a^2+b^2)y=0`

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Solution

y = eax. cos bx

`dy/dx=ae^(ax).cosbx-be^(ax).sinbx.........(i)`

`dy/dx=ay-be^(ax).sinbx`

`(d^2y)/(dx^2)=ady/dx-b(ae^(ax).sinbx+be^(ax).cosbx)`

`(d^2y)/(dx^2)=ady/dx-abe^(ax).sinbx-b^2e^(ax).cosbx`

`(d^2y)/(dx^2)=ady/dx-a(ay-dy/dx)-b^2y ` [Substituting beax sinbx from(i)]

`(d^2y)/(dx^2)=ady/dx-a^2y+ady/dx-b^2y`

`therefore (d^2y)/(dx^2)-2ady/dx+(a^2+b^2)y=0`

Hence Proved

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