Question

Find the general solution of the following nonhomogeneous second order differential equation: y" - 4y = e^2x

Answer 1

Solution is
`y=c_(1)e^(2x)+c_(2)e^(-2x)+(1)/(4)xe^(2x)`

Explanation:

the given equation y''-4y
`=e^(2x)` can be written as

`D^(2)y-4y=e^(2x)\\\\(D^(2)-4)y=e^(2x)\\\\`

The Complementary function thus becomes

y=c_{1}e^{m_{1}x}+c_{2}e^{m_{2}x}

where
`m_(1) , m_(2)` are the roots of the
`D^(2)-4`

The roots of
`D^(2)-4` are +2,-2 Thus the comlementary function becomes

`y=c_(1)e^(2x)+c_(2)e^(-2x)`

here
`c_(1),c_(2)` are arbitary constants

Now the Particular Integral becomes using standard formula

`y=(e^(ax))/(f(D))\\\\y=(e^(ax))/(f(a)) (f(a)\\eq 0)\\\\y=x(e^(ax))/(f'(a))(f(a)=0)`

`y=(e^(2x))/(D^(2)-4)\\\\y=(e^(2x))/((D+2)(D-2))\\\\y=(1)/(D-2)* (e^(2x))/(2+2)\\\\y=(1)/(4)* (e^(2x))/(D-2)\\\\y=(1)/(4)xe^(2x)`

Hence the solution is = Complementary function + Particular Integral

Thus Solution becomes
`y=c_(1)e^(2x)+c_(2)e^(-2x)+(1)/(4)xe^(2x)`