B5 Find the general solution of x 2x 4 6y et y x 3y etS

B5

Find the general solution of x\' = 2x 4- 6y + e^t y\' = x + 3y - e^t.

Solution

Inhomogeneous equation is:

x\'=2(x+3y)

y\'=x+3y

x\'+3y\'=5(x+3y)

(x+3y)\'=5(x+3y)

Integrating gives

x+3y= a exp(5t)

x=-3y+a exp(5t)

x\'=2 a exp(5t)+e^t

Integrating gives

x= 2a/5 exp(5t)+e^t+b

y\'= exp(5t)-e^t

y=exp(5t) /5-e^t+c