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Find a family of solutions of an underdetermined differential equation.

Posted 4 months ago
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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?

POSTED BY: Richard Jensen
3 Replies

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
POSTED BY: Hans Dolhaine
Posted 4 months ago

Very helpful. Thank you.

POSTED BY: Richard Jensen

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.

POSTED BY: Hans Dolhaine
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