Solve system of differential equations $$\begin{cases} x'(t)=2y(t)-x(t)+1 \\ y'(t)=3y(t)-2x(t). \end{cases} $$

My solution:

First, I find the characteristic values $$\begin{vmatrix} 3-\lambda & -2 \\ 2 & -1-\lambda \end{vmatrix}=-(3-\lambda)(1+\lambda)+4=0\Rightarrow \lambda_{1,2}=1$$ next, eigenvectors $$\begin{cases} 2x_1-2x_2=0 \\ 2x_1-2x_2=0 \end{cases}\Rightarrow x_1=1, \ x_2=1 \Rightarrow P_1=\begin{pmatrix} 1 \\ 1 \end{pmatrix}$$ $$\begin{cases} 2x_1-2x_2=1 \\ 2x_1-2x_2=1 \end{cases}\Rightarrow x_1=2, \ x_2=3\Rightarrow P_2=\begin{pmatrix} 2 \\ 3 \end{pmatrix}$$ This means that $$y_h=C_1e^tP_1+C_2e^t(tP_1+P_2)$$ Since our system is nonhomogeneous, we also need to find $y_*$, the particular solution. I would appreciate some information about the method.

  • $\begingroup$ You have a first order system with a polynomial of degree $0$ as the forcing. This motivates assuming that the particular solution is a polynomial of degree $1$. Now identify the (four) coefficients. $\endgroup$ – Ian Dec 10 '14 at 15:12

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