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Use the Laplace transform to convert the ODE to an algebraic equation

Use an integrating factor

Assume a solution that is the product of functions, each in a single variable, and separate

Use substitution and change to linear form

Solve for eigenvalues and eigenfunctions of a self-adjoint differential operator

Integrate the exact equation with respect to $x$ and $y$

Use Fourier transform to solve PDEs with non-periodic or infinite boundary conditions

Solve the characteristic equation

Use substitution $y = vx$ and separation of variables

Solve using characteristic curves, which are solutions to a related system of ODEs

Find new differential equations representing the trajectories

No general method; may use perturbation theory, numerical methods, or seek particular solutions

Expand the functions in terms of Fourier series

Solve via eigenvalues and eigenvectors or by diagonalization

Construct a solution for inhomogeneous boundary value problems

Use the method of undetermined coefficients or variation of parameters

Solve using Bessel functions of the first and second kind, $J_n$ and $Y_n$

Solve using Legendre polynomials

Separate variables and integrate both sides