Solve the differential equation dy/dx=xy+y^2

Most ODE's are analytically unsolvable.
It might be there exists a trick to manage this one, but off-hand, I can't help you.

 

The answer can not be expressed in terms of elementary functions. Although with the use of the quite commonly used Erfi function there is a fairly simple representation of the answer, why don't you show us what you have done so far.

 

It's a Bernoulli equation, you can solve it by Leibniz substitution

[tex]v=y^{(1-n)}[/tex]

 

Alright, this is how I see it starting from:

[tex]d\left[e^{x^2/2}z\right]=-e^{x^2/2}[/tex]

Integrating:

[tex]\int_{x_0,z_0}^{x,z} d\left[e^{x^2/2}z\right]=-\int_{x_0}^x e^{x^2/2}dx[/tex]

Yielding:

[tex]z=Ke^{-x^2/2}-e^{-x^2/2}\int_{x_0}^x e^{t^2/2}dt[/tex]

Noting that:


[tex]\int_{x_0}^x f(t)dt=K+\int_0^x f(t)dt[/tex]

We can write the above expression as:

[tex]z(y)=Ce^{-x^2/2}-e^{-x^2/2}\int_0^x e^{t^2/2}dt[/tex]

Letting [itex]u=t/\sqrt{2}[/itex] we obtain:

[tex]z(y)=Ce^{-x^2/2}-\sqrt{2}e^{-x^2/2}\int_0^{x/\sqrt{2}} e^{u^2}du\quad\tag{1}[/tex]

Now, it just so happens that:

[tex]Erfi[x]=\frac{Erf[ix]}{i}[/tex]

and:

[tex]Erf[ix]=\frac{2i}{\sqrt{\pi}}\int_0^{x} e^{t^2}dt[/tex]

so the i's cancel and we're left with:

[tex]Erfi[x]=\frac{2}{\sqrt{\pi}}\int_0^{x} e^{t^2}dt[/tex]

or:

[tex]\int_0^{x} e^{t^2}dt=\frac{\sqrt{\pi}}{2}Erfi[x][/tex]

Substituting that into (1) we get what Mathematica reports:

[tex]z(y)=Ce^{-x^2/2}-e^{-x^2/2}\sqrt{\frac{\pi}{2}}
Erfi\left[\frac{x}{\sqrt{2}}\right][/tex]

But z is 1/y so then 1 over all that stuf is the answer (if I didn't make any mistakes). How about a plot?

Also, I'll quote someone in here:

"Equal rights for special functions"

That means, treat Erfi[x] in the same way as Sin[x]. You wouldn't mind if the answer was:

[tex]z(x)=Ce^{-x^2/2}Sin[x][/tex]

would you? Same dif.

 

Oh yea, back-substitute everything into the ODE in Mathematica to make sure the answer is right. We can do that . . . like Gauss and the rest of them wouldn't have used Mathematica if they had it back then. Be for real.

 

Well Erfi[x] is called the imaginary Error function and Erf[x] is the error function. Check out Mathworld for details. But just any valid expression can hold for a new function. There's tons of them. For example, if I have some irreducible integral of the form:

[tex]2e^c\int_0^{\sqrt{ln(x)-c}} e^{t^2}dt[/tex]

I can call it Sal[x] such that:

[tex]Sal[x]=2e^c\int_0^{\sqrt{ln(x)-c}} e^{t^2}dt[/tex]

and further:

[tex]\frac{dSal[x]}{dx}=\frac{1}{\sqrt{ln(x)+c}}[/tex]


Same dif as Sin[x] as far as I'm concerned.