This is a first order differential equation:
$$
\frac{df_1}{dx} + \frac{(f_1)^2}{h^2} – \frac{2m}{h^2} \lambda \delta(x-pa)=-\frac{2mE_1}{h^2}
$$
Where, h, $\lambda $ and $E_1$ are constants and and pa lies in [0,a] as 0<p<1 .
I have not been taught how to handle differential equations with a Dirac Delta function in it. Moreover, this is a non linear one. I came across this in a research paper and the answer is given but the method to solve it isn't. I have tried learning to use Laplace transform to solve this, but got stuck again because I didn't know how to do Laplace transform of the second term of the equation.
Any help will be appreciated. Please, help me out.
The answer is:
$f_1=√2mE_1[cot (\frac{√2mE_1}{h} (x-b))]$
Where b is constant of integration
P.s.: I know this might be rude but please don't vote this as a homework question because it isn't one. If you can't help just ignore.
Best Answer
Using the notation in my comment, for $x\ne0$ the equation
$$g'+g^2=\pm1$$ is separable and the solution will be $g$ as the tangent or hyperbolic tangent of $x$, with constants.
The constants can differ on the left and on the right, and by introducing a discontinuity of height $a$, you will retrieve the Dirac Delta.
The solution can be like
$$x<0\to g(x)=-\tan(x+r),\\x>0\to g(x)= -\tan(x+s)$$
and there is a unit step at $x=0$ if $\tan(r)-\tan(s)=1.$