# Dsolve cannot solve this nonlinear ODE system

when I run this, I get "Supplied equations not differential or integral equations of the given functions" although the solution for this initial conditions is x = [sin(t),0,cos(t),0]. Why can't Dsolve solve this problem? Is there any way to get the general solution for this problem?

B = 0.5;
c = 10;
eqns = {x1'[t] == x3[t],
x2'[t] == x4[t],
x3'[t] == -x1[t] +
B*(-1 + x1[t]^2 + x2[t]^2 + x3[t]^2 + x4[t]^2),
x4'[t] == -c^2*x2[t] +
B*(-1 + x1[t]^2 + x2[t]^2 + x3[t]^2 + x4[t]^2), x1[0] == 0,
x2[0] == 0, x3[0] == 1, x4[0] == 0};
sols = DSolve[eqns, {x1, x2, x3, x4}, t]

• you have typo. You wrote x[4]==0 in the initial conditions. It should be x4[0]==0 but DSolve can not solve it still. Since it is non-linear system of ode's which is much harder to solve analytically than linear set of odes'. Aug 9 at 7:33
• @Nasser yep crap! thanks for the correction Aug 9 at 7:38

you have typo. You wrote x[4]==0 in the initial conditions. It should be x4[0]==0 but DSolve can not solve it still. Since it is non-linear system of ode's which is much harder to solve analytically than linear set of odes'

B = 1/2;
c = 10;
ClearAll[x1, x2, x3, x4, t];
eqns = {x1'[t] == x3[t],
x2'[t] == x4[t],
x3'[t] == -x1[t] + B*(-1 + x1[t]^2 + x2[t]^2 + x3[t]^2 + x4[t]^2),
x4'[t] == -c^2*x2[t] +
B*(-1 + x1[t]^2 + x2[t]^2 + x3[t]^2 + x4[t]^2)
};
ic = {x1[0] == 0, x2[0] == 0, x3[0] == 1, x4[0] == 0};
DSolve[{eqns, ic}, {x1, x2, x3, x4}, t]


Here is numerical

sol = NDSolveValue[{eqns, ic}, {x1, x2, x3, x4}, {t, 0, 30}]


Plot[Evaluate[(sol[[#]][t]) & /@ Range[4]], {t, 0, 30},
GridLines -> Automatic, GridLinesStyle -> LightGray]


Or to see them more clearly (since the domain changes)

p = Table[
Plot[sol[[n]][t], {t, 0, 30}, GridLines -> Automatic,
GridLinesStyle -> LightGray, PlotLabel -> Row[{"sol ", n}]], {n,
1, 4}];
Grid[{p}]


So first and third solutions are dominant and second and fourth are minimal.

• FWIW the symbolic solution to the IVP is made from the sine, cosine, and zero functions: {eqns, ic} /. {x1 -> Sin, x3 -> Cos, x2 | x4 -> (0 &)} // Simplify Aug 9 at 13:01