Find the particular solution to y5y6y17te2tSolutionFirst let

Find the particular solution to y\'\'+5y\'+6y=-17te^(2t)

Solution

First let\'s check the complimentary solution ie solution to homogeneous equation

y\'\'+5y\'+6y=0

Assume:y=e^{kx},substituting gives:

k^2+5k+6=0

k=-2,-3

Hence solutions of homogeneous ODE are: e^{-2t} and e^{-3t}

Since the inhomogeneous part is not solution to the ODE we can assume the form of particular solution based on the inhomogeneous part:

yp=e^{2t}(A+Bt)

yp\'=e^{2t}B+2e^{2t}(A+Bt)=e^{2t}(2A+B+2Bt)

yp\'\'=e^{2t}2B+2e^{2t}(2A+B+2Bt)=e^{2t}(4A+4B+4Bt)

Substituting gives:

4A+4B+4Bt+5(2A+B+2Bt)+6(A+Bt)=-17t

Comparing constant terms and coefficients of t gives:

4A+4B+10A+5B+6A=0   ,Hence, 20A+9B=0

4Bt+10Bt+6Bt=-17t

So, B=-17/20

A=153/400

So particular solution is:

e^{2t}(153/400-17t/20)