I would really appreciate it if someone could show me the steps to how to solve such a differential equation.
$\displaystyle \dot{x}=32x-64y$
$\displaystyle \dot{y}=-3x+48y$
$\displaystyle x(0)=-64$
$\displaystyle y(0)=4$
Thank you very much.
I would really appreciate it if someone could show me the steps to how to solve such a differential equation.
$\displaystyle \dot{x}=32x-64y$
$\displaystyle \dot{y}=-3x+48y$
$\displaystyle x(0)=-64$
$\displaystyle y(0)=4$
Thank you very much.
There are at least a couple of options:
1. Use the eigenvalue/eigenvector diagonalization approach.
2. Eliminate one variable to get a second-order DE.
You can check those out at Chris's DE tutorial. Is a method specified for this problem, or can you use any method?
$\displaystyle \displaystyle \frac{dx}{dt} = 32x - 64y$
$\displaystyle \displaystyle \frac{dy}{dt} = -3x + 48y$.
$\displaystyle \displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
$\displaystyle \displaystyle \frac{dy}{dx} = \frac{-3x + 48y}{32x - 64y}$
$\displaystyle \displaystyle (32x - 64y)\,dy = (-3x + 48y)\,dx$
$\displaystyle \displaystyle (32x - 64y)\,dy + (3x - 48y)\,dx = 0$.
Now you can probably find an integrating factor to turn this into an exact equation.
First write the system as a matrix.
Let $\displaystyle \mathbf{X}=\begin{pmatrix} x(t) \\ y(t) \end{pmatrix}$
And $\displaystyle A=\begin{pmatrix} 32 & -64 \\ -3 & 48 \end{pmatrix}$
Now we have $\displaystyle \dot{\mathbf{X}}=A\mathbf{X}$
Here is a link on how to solve a matrix system ODE. Scroll down a bit to find the computation of eigenvalues and eigenvectors section. They also have examples of this type of problem.
Eigenvalues and Eigenvectors Technique