Solve the given differential equation by undetermined coeffi

Solve the given differential equation by undetermined coefficients. Y\'\' - 10y\' + 25y = 35x + 5 y(x) = __________

Solution

First we solve the associated homogneous ode

y\'\'-10y\'+25y=0

It is a linear,homogeneous iwth constant coefficients. SO, y=exp(kx)

Substituting gives

k^2-10k+25=0

k=5,repeated root

yh=exp(5x)(Ax+B)

Now for the particular solution we guess, yp=Ax+B

Susbtituting gives

-10A+25(Ax+B)=35x+5

25Ax-10A+B=35x+5

A=7/5

-10A+B=5

-14+B=5

B=19

yp=7x/5+19

y=yh+yp=exp(5x)(Ax+B)+7x/5+19