Question

Solve the following systems of equations

2dx/dt+dy/dt-3x-y=t

dx/dt+dy/dt-4x-y=e^t

Answer 1

$$2x'+y'-3x-y=t$$

$$x'+y'-4x-y=e^t$$

$$y'=-2x'+3x+y+t$$

$$y'=-x'+4x+y+e^t$$

$$-2x'+3x+y+t=y'=-x'+4x+y+e^t$$

$$x'+x=t-e^t$$

$$\mu(t)=e^t$$

$$e^tx'+e^tx=e^tt-e^te^t$$

$$(e^tx)'=e^tt-e^{2t}$$

Integrate

$$\int d(e^tx)=\int(e^tt-e^{2t})dt$$

$$\int e^ttdt=te^t-e^t+C_3$$

$$e^tx=te^t-e^t-\dfrac{1}{2}e^{2t}+C_1$$

$$x(t)=t-1-\dfrac{1}{2}e^t+C_1e^{-t}$$

$$x'=1-\dfrac{1}{2}e^t-C_1e^{-t}$$

$$2-e^t-2C_1e^{-t}+y'-3t+3+\dfrac{3}{2}e^t-3C_1e^{-t}-y=t$$

$$y'-y=4t-5-\dfrac{1}{2}e^t+5C_1e^{-t}$$

$$\mu(t)=e^{-t}$$

$$e^{-t}y'-e^{-t}y=4te^{-t}-5e^{-t}-\dfrac{1}{2}+5C_1e^{-2t}$$

$$(e^{-t}y)'=4te^{-t}-5e^{-t}-\dfrac{1}{2}+5C_1e^{-2t}$$

Integrate

$$\int d(e^{-t}y)=\int(4te^{-t}-5e^{-t}-\dfrac{1}{2}+5C_1e^{-2t})dt$$

$$e^{-t}y=-4te^{-t}-4e^{-t}+5e^{-t}-\dfrac{1}{2}t-\dfrac{5}{2}C_1e^{-2t}+C_2$$

$$y(t)=-4t+1-\dfrac{1}{2}te^{t}-\dfrac{5}{2}C_1e^{-t}+C_2e^{t}$$

$$x(t)=t-1-\dfrac{1}{2}e^t+C_1e^{-t}$$

$$y(t)=-4t+1-\dfrac{1}{2}te^{t}-\dfrac{5}{2}C_1e^{-t}+C_2e^{t}$$