Find a particular solution to the nonhomogeneous differentia

Find a particular solution to the nonhomogeneous differential equation y\" + 4y\' + 5y= 10x + 5e^-x y_p = Find the most general solution to the associated homogeneous differential equation Use c_1 and c_2 in your answer to denote arbitrary constants and enter them as c_1 and c_2 y_h Find the most general solution to the original nonhomogeneous differential equation Use c_1and c_2 in your answer to denote arbitrary constants y =

Solution

a.

Based on the inhomogeneous part the guess for particular solution is

yp=Ax+B+Ce^{-x}

Substituting gives

Ce^{-x}+4(A-Ce^{-x})+5(Ax+B+Ce^{-x})=10x+5e^{-x}

Comparing coefficients gives

2C=5, C=5/2

4A+B=0

5A=10 giving A=2

B=-8

yp=2x-8+5e^{-x}/2

b)

Here the associated homogeneous equation is

y\'\'+4y\'+5y=0

It is a linear homogeneous recurrence so solution must be of the form

y=exp(kx)

Substituting gives

k^2+4k+5=0

So

k=-2+i,-2-i

So ,

yh=e^{2x}(A sin(x)+B cos(x))

c.

y=yh+yp=2x-8+5e^{-x}/2+e^{2x}(A sin(x)+B cos(x))