Find the solutions to the intial value problems by using sol

Find the solutions to the intial value problems by using solutions of nonhomogeneous systems:

Find the solutions to the initial value problems: x\' = 4x - 6y + 10 y\' = x - y x(0) = 0, y(0) = 0

Solution

First we solve associated homogeneous system

x\'=4x-6y

y\'=x-y

From second equation

x=y\'+y

x\'=y\'\'+y\'=4x-6y=4(y\'+y)-6y

y\'\'+y\'=4y\'-2y

y\'\'-3y\'+2y=0

Let,y=exp(kt)

Substituting gives

k^2-3k+2=0

k=1,2

So,

y=A exp(t)+B exp(2t)

y\'=A exp(t)+2B exp(2t)

x=y+y\'=2A exp(t)+3B exp(2t)

x=2A exp(t)+3B exp(2t)

Now for particular solution be

xp=C,yp=D

Substituting gives

0=4C-6D+10

0=C-D ie C=D

0=4C-6C+10

C=5=D

So,

x=2A exp(t)+3B exp(2t)+5

y=A exp(t)+B exp(2t)+5

x(0)=2A+3B+5=0

y(0)=A+B+5=0

Solving gives:

A=-10,B=5