# Thread: the process of solving a differetial equation

1. ## the process of solving a differetial equation

Hi, im having some difficulties solving differential equations. It's not very well explained in my book.

Take for example the equation $\displaystyle dy/dx -2y=4x$
$\displaystyle y(0)=1$

The first step is to check what kind of function you are dealing with. It's a first degree funktion where the general solution is $\displaystyle y=ax+b$
or was it $\displaystyle C*e^{-ax}$

Anyway if somebody please could explain the steps involved i'd be grateful

As im lousy at math...

It's fun when you understand what you're doing but can be hell if you don't.

//Jones

2. This DE is a 1st order linear.

Find your integrating factor.

$\displaystyle \frac{dy}{dx}-2y=4x$

Integrating factor is $\displaystyle e^{\int{-2}dx}=e^{-2x}$

$\displaystyle \frac{d}{dx}[e^{-2x}y]=e^{-2x}\cdot{4x}$

Integrate both sides:

$\displaystyle e^{-2x}y=-(2x+1)e^{-2x}+C$

Solve for y:

$\displaystyle y=-2x-1+Ce^{2x}$

Use I.C. to find C:

$\displaystyle 1=-2x-1+Ce^{2(0)}$

C=2

$\displaystyle y=-2x-1+2e^{2x}$