Nonhomogeneous ODEs that can't be made homogeneous?

[TylerH]

Assuming knowledge of homogeneous ODEs and nonhomogeneous ODEs that can be made homogeneous (eg, y'-y=x), how does one solve those that cannot be made homogeneous (eg, y'-y=cosx, y''-xy'+y=0, cos(y'')+sin(y')=0)?

EDIT: Maybe "made homogeneous" is the wrong way to put it... By being able to be "made homogeneous," I mean it is possible to differentiate the right hand side to 0 so as to find the general form of the particular solution.

 

[3.1415926535]

There is no method to solve all ODEs but there are methods to solve some of them
All first order linear ODEs can be solved(e.g y'-y=cosx, y'+ye^x-lnx=2 )
All ODEs with constant coefficients can be solved(e.g. y'''+5y'-4y=2, 7y''+2y'=y)(assuming you can find the roots of an nth order polynomial for an nth order ODE)
Some ODEs with variable coefficients can be solved(e.g. y''+xy'+y=0, x^2y''-y=3)

Note that the solutions are not stricly elementary functions but they can at least be expressed as a power series or an integral
Regarding your examples, the first can be solved easily(using elementary functions) as well as the second(using Frobenius power series)
The third however, is not linear... and pretty much unsolvable

 

[Skrew]

Anything whichs derivative does not go to zero.

y'' + y = ln(x).

You have to use variation of parameters.