Use the method of undetermined coefficients to solve the giv

Use the method of undetermined coefficients to solve the given system. X\' = (1 3 3 1)X + (-4t^2 t + 3) X(t) =

Solution

Let, X=(u,v)^T

So

u\'=u+3v-4t^2

v\'=3u+v+t+3

First we solve homogeneous equations

u\'=u+3v

v\'=3u+v

Adding equations gives

(u+v)\'=4(u+v)

Integrating gives

u+v=A e^{4t}

Substracting equations gives

(u-v)\'=-2(u-v)

Integrating gives

u-v=B e^{-2t}

So,

u=A e^{4t} +B e^{-2t}

v=A e^{4t} - Be^{-2t}

Let guesses for particular solution be:

up = p1 t^2+q1 t+ r1

vp =p2 t^2+q2 t+ r2

Substituting gives

2p1 t+q1=(p1+3p2)t^2+(q1+3q2)t+(r1+3r2)-4t^2

p1+3p2=4

q1+3q2=2p1

r1+3r2=q1

2p2 t+q2=(p2+3p1)t^2+(q2+3q1)t+(r2+3r1)+4t+3

p2+3p1=0

q2+3q1+4=2p2

r2+3r1+3=q2

Solving gives:

p2=-1/2,p1=1/6

q1+3q2=2p1 , q2+3q1+4=2p2

q1+3q2=1/3

q2+3q1=-5

q2=3/4,q1=-23/12

r1+3r2=q1, r2+3r1+3=q2

r1+3r2=-23/12, r2+3r1+3=3/4

r1=-29/48, r2=-7/16

so we have the particular solution and this solves the differential equation