Second order ordinary differential equations I have a question about second-order ordinary differential equations. what is the proper method for solving these equations? And in a case, if we have a set of these equations with only one unknown function ($f(r)$) as the following form:
$
1) a1 f(x)+b2 f'(x)+c1 f''(x)+d1 f(x) f'(x)+e1 f'(x) f''(x)+h1 f(x) f''(x)=0\\
2) a2 f(x)+b3 f'(x)+c3 f''(x)+d2 f(x) f'(x)+e2 f'(x) f''(x)+h2 f(x) f''(x)=0\\
3) a3 f(x)+b3 f'(x)+c3 f''(x)+d3 f(x) f'(x)+e3 f'(x) f''(x)+h3 f(x) f''(x)=0\\
4) a4 f(x)+b4 f'(x)+c4 f''(x)+d4 f(x) f'(x)+e4 f'(x) f''(x)+h4 f(x) f''(x)=0\\
$
Can we solve that? And How can we understand if these equations have a solution?
Also, I have the following equations which is a more complicated case
\begin{equation}
\pmb{\text{eq1}=\frac{(a_1) l^2+r^2}{\left(l^2+r^2\right)^2}+\frac{(a_2) f'[r]^2}{8 f[r]^2}+\frac{-1+r
f'[r]}{\left(l^2+r^2\right) f[r]}=0}\\
\pmb{\text{eq2}=\frac{(a_3) l^2 f[r]}{l^2+r^2}+r f'[r]-\frac{(a_4) \left(l^2+r^2\right) f'[r]^2}{8 f[r]}+\frac{1}{2}
\left(l^2+r^2\right) f''[r]=0}
\end{equation}
What are your suggestions to solve these equations?
Thanks
 A: There is no "proper method" in general. What can be done is identify classes of equations and then use a variety of methods that are known to work for those classes. For example, linear equations with nice enough coefficents can be solved via Laplace transforms or, or homogenous scalar equations may be integrated with suitable variable transformations and then quadratures (like what you do for first order homogenous), in the case of Bessel type equations, you can calculate a point transformation that lets you transform the equation into a normal form. etc.
The most general methods however for integrating equations in general are either by identifying and studying certain Hilbert/Banach spaces associated with the equations or to use symmetries reduce to order of the equation (this is usually done through coordinate transformations). In general, you cannot find the exact form of a solution because it doesn't exist however so you just do casework for initial value problems.
In the case of your listed equation, since there is more than one equation you should start by looking for compatibility conditions. Second, there's also the algebra perspective of this, you can look for combinations of the equations that are nicer to deal with or perhaps a combination that yields a first order equation, see if you can solve this, and then substitute it into all the equations to verify that it works and then do all the specific casework associated with whatever conditions you have to have in order to integrate the equation. You might find that a solution however does not exist depending on the values of the coefficients.
