Numerical solution of a linear differential equation with non-constant coefficients

Question

I'm trying to solve the following linear differential equation with non-constant coefficients involving algebraic functions.

$\frac{d \phi}{d t}+\frac{ \left(\Delta ^2 \tau ^2-4 i \tau \sqrt{\Delta ^2+\left(\frac{t (\text{$\epsilon $1}-\text{$\epsilon $0})}{\tau }+\text{$\epsilon $0}\right)^2}+(\tau -2 t)^2 \left(\Delta ^2+\left(\frac{t (\text{$\epsilon $1}-\text{$\epsilon $0})}{\tau }+\text{$\epsilon $0}\right)^2\right)\right)\text{$\phi $}(t)}{4 \tau ^2}=0$

where, $\Delta$, $\tau$, $\epsilon_0$, and $\epsilon_1$ are parameters.

Analytically this seems difficult to solve. Are there any numerical methods to solve this differential equation? Any guidance on this would be much appreciated.

Answer 1

$$-4\tau^2\log\phi=\int\left(\Delta ^2 \tau ^2-4 i \tau \sqrt{\Delta ^2+\left(\frac{t (\epsilon_1-\epsilon_0)}{\tau }+\epsilon_0\right)^2}+(\tau -2 t)^2 \left(\Delta ^2+\left(\frac{t(\epsilon_1-\epsilon_0)}{\tau }+\epsilon_0\right)^2\right)\right)dt.$$ The integral is manageable.

Answer 2

The first-order linear ordinary differential equation you want to solve, is:

$$\text{y}'\left(t\right)+\text{p}\left(t\right)\text{y}\left(t\right)=0\space\Longleftrightarrow\space\frac{\text{y}'\left(t\right)}{\text{y}\left(t\right)}=-\text{p}\left(t\right)\tag1$$

Integrate both sides:

$$\int\frac{\text{y}'\left(t\right)}{\text{y}\left(t\right)}\space\text{d}t=\int-\text{p}\left(t\right)\space\text{d}t\tag2$$

And this gives:

$$\ln\left|\text{y}\left(t\right)\right|=-\int\text{p}\left(t\right)\space\text{d}t\space\Longleftrightarrow\space\left|\text{y}\left(t\right)\right|=\exp\left\{-\int\text{p}\left(t\right)\space\text{d}t\right\}\tag3$$