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.

- Apr 4th 2011, 05:07 PMiPodNeed 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.

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

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

$\displaystyle x(0)=-64$

$\displaystyle y(0)=4$

Thank you very much. - Apr 4th 2011, 05:45 PMAckbeet
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, 06:17 PMiPod
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, 06:24 PMProve It
$\displaystyle \displaystyle \frac{dx}{dt} = 32x - 64y$

$\displaystyle \displaystyle \frac{dy}{dt} = -3x + 48y$.

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

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

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

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

Now you can probably find an integrating factor to turn this into an exact equation. - Apr 4th 2011, 06:29 PMiPod
:| you are amazing

but ive forgotten how to find the integrating factor for exact equations, could you please remind me :) - Apr 4th 2011, 07:00 PMProve It
- Apr 4th 2011, 07:21 PMTheEmptySet
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