enter image description hereWhile experimenting with diodes I have come across the following differential equation:
Let $ a_1 , a_2 , a_3, a_4,$ and $K$ be real numbers greater than zero.
$K = a_1 y' + a_2 y + a_3 ln(a_4y')$
$y(0)=0$
Does anyone have any advice for how to tackle that equation? That natural log is really tripping me up.
Thank you in advance for your help.
Considering the equation $$K = a_1 y' + a_2 y + a_3 \log(a_4y')$$ where $y$ is a function of $x$.
First, solve for $y'$; the solution is given in terms of Lambert function $$y'=\frac{{a_3} }{{a_1}}\,\,W\left(\frac{{a_1} }{{a_3} {a_4}}e^{\frac{K-a_2 y}{{a_3}}}\right)=\frac 1{x'}$$ Now, integrate with respect to $y$ (this is not the most funny part, but a CAS did it) and get $$x+C =\frac {a_1}{a_2}\left(\frac 1z-\log(z) \right)\qquad \text{with}\qquad z=W\left(\frac{{a_1} }{{a_3} {a_4}}e^{\frac{K-a_2 y}{{a_3}}}\right)$$