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I have an exercise to first model then solve an ODE system, which I hope I have done correctly, and then solve the problem $$\dot{\overrightarrow{u}}=A\overrightarrow{u}+\overrightarrow{b}$$ using the sytem I found.

My problem is that I'm not really sure how to approach the problem. $\overrightarrow {b} $ is not a set of functions, of which I have found examples, but just a normal vector. I know it's not an initial value problem but I'm not sure how to handle it.

Here is the problem:

$$\dot{\overrightarrow{u}}=\begin{bmatrix}-\frac{1}{10} & 0\\ \frac {1}{10} & -\frac{1}{10} \end{bmatrix}\cdot\overrightarrow{u}+\begin{bmatrix}1\\-1 \end{bmatrix} $$

with

$$u_{1}=e^{t}\left(\begin{array}{c}0\\ 1\end{array}\right)$$ $$u_{2}=e^{t}\left(\begin{array}{c}1\\ 1+t\end{array}\right)$$

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1 Answer 1

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You can use the method of undetermined coefficients or educated guessing to find a particular solution. Since $0$ is not an eigenvalue of $A$ and $\vec b$ is constant, look for a particular solution of the form $$ \vec u=\vec{u_p},\quad\vec{u_p}\text{ a constant vector.} $$ Substituting in the equation we get $$ A\,\vec{u_p}=\vec b\implies\vec{u_p}=A^{-1}\vec b. $$ The general solution is then $$ \vec u=C_1\,\vec{u_1}+C_2\,\vec{u_2}+\vec{u_p}. $$

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