- #1
heman
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How to solve the differential equation
dy/dx=xy+y^2
In my case this leads to an integral which is unsolvable!
dy/dx=xy+y^2
In my case this leads to an integral which is unsolvable!
heman said:That's exactly what i have done,but later it is unsolvable,or rather solution is not clear to me.
saltydog said: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.
heman said:Thanks Salty
Everything is fine and well for me in yours solution!
Could you please throw some light on this New Emergent function!
I came across this first time!
A differential equation is a mathematical equation that relates the rate of change of a function to the function itself. It is used to model many natural phenomena and is an important tool in science and engineering.
To solve a differential equation, you need to find a function that satisfies the equation. This can be done through various methods such as separation of variables, substitution, or using an integrating factor.
The notation dy/dx represents the derivative of the function y with respect to x. It represents the rate of change of y with respect to x.
This differential equation can be solved using the method of separation of variables. First, we can rearrange the equation to get dy/dx=y(x+y). Then, we can separate the variables to get 1/y dy = (x+y)dx. By integrating both sides, we get ln|y|=x^2/2 + xy + C. Finally, we can solve for y to get y(x)=Ce^(x^2/2+xy).
Differential equations are used to model many real-world phenomena in fields such as physics, chemistry, biology, engineering, and economics. They can be used to predict the behavior of systems over time and are essential in understanding complex systems.