# Particular second order differential equation with variable coefficients

I would like to solve this second order differential equation with variable coefficients :

$a(x)*y^{''}(x)+b(x)*y(x) = f(x)$

The coefficients can be given by

• linear equations : $a(x) = ax + b$, $b(x) = cx + d$ and $f(x) = k*b(x)$

with a, b, c, d and k constant

• real exponential functions

• combination of both

could you advice me reference texts where a method to solve such problems is described. I couldn't find any ...

## 1 Answer

$$(ax+b)y'' + (cx+d)y = k(cx+d)$$ firs thing $$(ax+b)y'' + (cx+d)(y-k) = 0$$ so change of variable $v = y-k$

$$(ax+b)v'' + (cx+d)v = 0$$ next thing multiply by $v'$ we find

$$(ax+b)v''v' + (cx+d)vv' = 0$$

Then using the sub $t = ax+ b$ we can obtain $$\frac{d^2v}{dt^2} +\left(\lambda_1 + \frac{\lambda_2}{t}\right)v = 0$$ which you could try and solve using power series? maybe

As for more general cases $$v'' + \frac{b(x)}{a(x)}v = 0$$ where $v = y - k$ which you have to try and solve on a case by case bases

If $$b(x) = \beta \mathrm{e}^{\gamma_2 x}\\ a(x) = \alpha \mathrm{e}^{\gamma_1 x}\\$$ then the solution is $$y(x) = c_1J_{0}\left(\frac{2\sqrt{\frac{\beta}{\alpha}\mathrm{e}^{\left(\gamma_2-\gamma_1\right)x}}}{\left(\gamma_2-\gamma_1\right)}\right)+ 2c_2Y_{0}\left(\frac{2\sqrt{\frac{\beta}{\alpha}\mathrm{e}^{\left(\gamma_2-\gamma_1\right)x}}}{\left(\gamma_2-\gamma_1\right)}\right) + k$$

you can try to check further the others.