Find a family of solutions of an underdetermined differential equation.

Questions
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1

The differential equation

g[x]s''[x] + 2g[x]s[x] + g''[x]s[x] == g[x]/2

has two independent variables and is thus underdetermined. Is there any way for Mathematica to produce a complete family of possible solutions?

2

Your differential equation is

deq = g[x] s''[x] + 2 g[x] s[x] + g''[x] s[x] == g[x]/2

Now make the following rearrangements

de1 = #/g[x] & /@ deq // Expand
de2 = # - 2 s[x] - s''[x] & /@ de1
de3 = #/s[x] & /@ de2

so the functions g and s are separated and you may set each side of the resulting equation de3 to an arbitrary constant or an arbitrary convenient function and solve the resulting differential equations.

For example

fg = g[x] /. DSolve[D[g[x], x, x]/g[x] == a, g[x], x][[1]]

fs = s[x] /.  DSolve[(1/2 - 2 s[x] - D[s[x], x, x])/s[x] == a, s[x], x][[1, 1]]

and your differential equation again

2 fg fs + fs D[fg, x, x] + fg D[fs, x, x] - fg/2 // FullSimplify
3

Very helpful. Thank you.

4

Note that you should change the integration-constants C[1] and C[2] in fs to K[1] and K[2] (or any others), that doesn’t change the result. So you have 5 parameters: a, C[1], C[2], K[1] and K[2] in this sort of solution.

I checked whether replacing a by a*x works also. In my system (Mathematica V7) it does, albeit the result is quite complicated. So you get quite an amount of possible solutions.