# 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?
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Posted 4 months ago
 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 4 months ago
Posted 4 months ago
 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.
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