Question

Use theorem 9.3.13 to solve the initial value problem

$\frac{\mathbf{d}\overrightarrow{\mathbf{x}}}{\mathbf{d}\mathbf{t}}\mathbf{=}\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)\overrightarrow{\mathbf{x}}$with$\overrightarrow{\mathbf{x}}\left(0\right)\mathbf{=}\left[\begin{array}{c}1\\ -1\end{array}\right]$ .

Hint: Find first x2(t) and then x1(t).

Short answer

The solution is$e^t\left[\begin{array}{c}1-2t\\ -1\end{array}\right]$

Step-by-step solution

Definition of first order linear differential equation.

Consider the differential equation $\mathit{f}\mathbf{\text{'}}\left(t\right)\mathbf{-}\mathit{a}\mathit{f}\left(t\right)\mathbf{=}\mathit{g}\left(t\right)$where g(t)is a smooth function and 'a'is a constant. Then the general solution will be$\mathit{f}\left(t\right)\mathbf{=}{\mathit{e}}^{\mathbf{a}\mathbf{t}}\mathbf{\int }{\mathit{e}}^{\mathbf{-}\mathbf{a}\mathbf{t}}\mathit{g}\left(t\right)\mathit{d}\mathit{t}$.

Determination of the solution.

Consider the differential equation as follows.

$$\begin{gathered} \frac{d\overrightarrow{x}}{dt}=\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)\overrightarrow{x} \\ A=\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right) \\ A=1 \\ f\text{'}\left(t\right)-1f\left(t\right)=0 \end{gathered}$$

Now, the differential equation is in the form as follows.

$f\text{'}\left(t\right)-af\left(t\right)=g\left(t\right)$, where g(t) is a smooth function, then the general solution will be as follows.

$f\left(t\right)=e^{at}\int e^{-at}g\left(t\right)dt$

Compute the calculation of the solution.

Substitute the value 0 for g(t) and 1 for a in$f\left(t\right)=e^{at}\int e^{-at}g\left(t\right)dt$as follows.

$$\begin{gathered} f\left(t\right)=e^{at}\int e^{-at}g\left(t\right)dt \\ f\left(t\right)=e^t\int e^{-t}\times 0\times dt \\ f\left(t\right)=e^t\int 0\times dt \\ f\left(t\right)=e^t.C \end{gathered}$$

Here C is constant

Also it can be written as follows.

$C=\left[\begin{array}{c}1-2t\\ -1\end{array}\right]$

The solution is$e^t\left[\begin{array}{c}1-2t\\ -1\end{array}\right]$