# First order linear ODE solution

• Apr 2nd 2012, 03:44 AM
Jesssa
First order linear ODE solution
Hey,

So I've been trying to solve this ode,

I got stuck at the point where I end up integrating ex^2dx

The ODE is

y'=1-2xy with y(0)=0

I double checked with wolframalpha but it spits out a solution involving the complex error function (which I have never studied before)

Is there some other way to solve this ODE?
• Apr 2nd 2012, 04:59 AM
Prove It
Re: First order linear ODE solution
Quote:

Originally Posted by Jesssa
Hey,

So I've been trying to solve this ode,

I got stuck at the point where I end up integrating ex^2dx

The ODE is

y'=1-2xy with y(0)=0

I double checked with wolframalpha but it spits out a solution involving the complex error function (which I have never studied before)

Is there some other way to solve this ODE?

\displaystyle \displaystyle \begin{align*} e^{x^2} \end{align*}, while the integral does exist, it can not be written in terms of elementary functions. So you will be able to leave your solution in terms of \displaystyle \displaystyle \begin{align*} \int{e^{x^2}\,dx} \end{align*}.
• Apr 2nd 2012, 05:27 AM
Jesssa
Re: First order linear ODE solution
Oh cool thanks!

Hey, I also have this list of other questions to practice on using afew odes including the one above,

For the one above I've been having trouble showing b,

http://img194.imageshack.us/img194/6343/asdahi.jpg

From reading them it looks like b implies c aswell is that right?

If all approximations exist on |x|≤1/2 then they would converge to the solution right?

And the first theorm in my notes states that if |x|≤h≤a then there exists a unique solution y=phi, where h = Min(a, b/M) and |f(x,y)|=|1-2xy|≤M

But I've been having trouble showing b, would you be able to point me in the right direction?

I worked out part d) for one of the other odes following an example in my notes but for this one i get, using the below relation

$\displaystyle |{{\phi }_{n}}-\phi |\,\le \frac{M}{k}\frac{{{(kh)}^{n+1}}}{(n+1)!}{{e}^{kh}} \,\,\,\,\,\,where\,\,\,\left| \frac{\partial f}{\partial y} \right|\le k$

|f(x,y)|=|1-2xy|≤1 + 2 (1/2) 1 = 2 = M
|df(x,y)/dy|=|1-2x|≤2=k

and h = 1/2

Subbing it all into the equation i get ≤0.11

Is there something I am doing incorrectly? I took this approach before and got <0.01