# Need help with how to solve a certain type of differential equation

• Apr 4th 2011, 06:07 PM
iPod
Need help with how to solve a certain type of differential equation
I would really appreciate it if someone could show me the steps to how to solve such a differential equation.

$\dot{x}=32x-64y$

$\dot{y}=-3x+48y$

$x(0)=-64$

$y(0)=4$

Thank you very much.
• Apr 4th 2011, 06:45 PM
Ackbeet
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?
• Apr 4th 2011, 07:17 PM
iPod
its through the method of finding eigenvalues/vectors however i have no idea how to do so - if theres a tutorial online, id be very greatful
• Apr 4th 2011, 07:24 PM
Prove It
$\displaystyle \frac{dx}{dt} = 32x - 64y$
$\displaystyle \frac{dy}{dt} = -3x + 48y$.

$\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

$\displaystyle \frac{dy}{dx} = \frac{-3x + 48y}{32x - 64y}$

$\displaystyle (32x - 64y)\,dy = (-3x + 48y)\,dx$

$\displaystyle (32x - 64y)\,dy + (3x - 48y)\,dx = 0$.

Now you can probably find an integrating factor to turn this into an exact equation.
• Apr 4th 2011, 07:29 PM
iPod
:| you are amazing

but ive forgotten how to find the integrating factor for exact equations, could you please remind me :)
• Apr 4th 2011, 08:00 PM
Prove It
• Apr 4th 2011, 08:21 PM
TheEmptySet
Quote:

Originally Posted by iPod
I would really appreciate it if someone could show me the steps to how to solve such a differential equation.

$\dot{x}=32x-64y$

$\dot{y}=-3x+48y$

$x(0)=-64$

$y(0)=4$

Thank you very much.

First write the system as a matrix.

Let $\mathbf{X}=\begin{pmatrix} x(t) \\ y(t) \end{pmatrix}$

And $A=\begin{pmatrix} 32 & -64 \\ -3 & 48 \end{pmatrix}$

Now we have $\dot{\mathbf{X}}=A\mathbf{X}$

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