Find a particular solution of the differential equation 54y2

Find a particular solution of the differential equation 54y+2y+y = 3xe^2x using the Method of Undetermined Coefficients (primes indicate derivatives with respect to x).

yp = ?

Solution

Since the inhomogeneous part is

3xe^{2x} so the guess for particular solution is

yp=(Ax+B)e^{2x}

yp\'=(2Ax+A+2B)e^{2x}

yp\'\'=(4Ax+4A+4B)e^{2x}

Substituting gives

-54(4Ax+4A+4B)e^{2x}+2(2Ax+A+2B)e^{2x}+(Ax+B)e^{2x}=3xe^{2x}

-216Ax-216A-216B+4Ax+2A+4B+Ax+B=3x

-211A=3   , A=-3/211

-214A-211B=0

B=-214A/211=642/211^2=642/44521

yp=(-3x/211+642/44521)e^{2x}