I am currently working on a differential equation problem and could use some help in solving it. The problem is to find the general solution of the differential equation:
$$ \mathbf{x}^{\prime}=A \mathbf{x} $$
where the matrix $A$ has eigenvalues 2 and 1 with the corresponding eigenvectors $[2,-3]^{T}$ and $[-3,5]^{T}$.
I have attempted to solve the problem by first finding the matrix $P$ that diagonalizes $A$, which can be found by stacking the eigenvectors into the columns of the matrix $P$ and calculating the inverse of $P$:
$$ P=\begin{bmatrix} 2 & -3 \\ -3 & 5 \end{bmatrix},\quad P^{-1}=\frac{1}{16}\begin{bmatrix} 5 & 3 \\ 3 & 2 \end{bmatrix} $$
Then, I can diagonalize $A$ as:
$$ A=PDP^{-1}=\begin{bmatrix} 2 & -3 \\ -3 & 5 \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} \frac{1}{16}\begin{bmatrix} 5 & 3 \\ 3 & 2 \end{bmatrix} =\frac{1}{16}\begin{bmatrix} 1 & -3 \\ -3 & 5 \end{bmatrix} \begin{bmatrix} 5 & 3 \\ 3 & 2 \end{bmatrix} =\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} $$
Thus, the solution to the differential equation is given by:
$$ \mathbf{x}(t)=e^{At}\mathbf{c}=Pe^{Dt}P^{-1}\mathbf{c} $$
where $\mathbf{c}$ is the constant vector determined by the initial conditions.
However, I am not entirely confident in my solution and would appreciate any feedback or guidance on whether my approach and solution are correct.
Thank you in advance for your help!
Assuming that the matrix $A$ is $2\times 2$ and you have found $2$ distinct eigenvalues $\lambda_1,\lambda_2$ corresponding to two $2$ linearly independent eigenvectors $v_1,v_2$, you have found a basis of eigenvectors.
This means that the matrix $A$ is diagonalisable. However there is no need to try and diagonalise the matrix yourself in order to find the solution to the system. In fact you've made several mistakes while doing so. It is best to avoid this since the solution the the equation is simply
$$ \alpha_1 x_1(t) + \alpha_2x_2(t)$$ where $$x_i(t) = e^{\lambda_i t}v_i$$ verifies $x_i'(t) = Ax_i$ and $\alpha_1,\alpha_2$ are constants defined by the initial conditions.