# Thread: First order linear ODE solution

1. ## 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?

2. ## Re: First order linear ODE solution

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 \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 \begin{align*} \int{e^{x^2}\,dx} \end{align*}.

3. ## 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,

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

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

|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