# Why can't MMA solve this system of differential equations?

This system of differential equations can be solved in order to
obtain the expression of x[t] y[t] z[t]

DSolve[{x''[t] + x[t] == a, y''[t] == x[t]^2 - y[t],
z[t] == 2 x[t]*y[t] - z''[t]}, {x, y, z}, t]


But why can' t MMA solve this system of differential equations? We already know the solutions of the following differential equations

DSolve[{x''[t] + x[t] == a, y''[t] == x[t]^2 - y[t],
z[t] == 2 x[t]*y[t] - z''[t], x == (1 + E) a, x' == 0,
y == 0, y' == 0, z == 0, z' == 0}, {x[t], y[t], z[t]}, t]

x[t] -> a + a E Cos[t]
y[t] -> -(1/6)
a^2 (2 (3 + E^2) Cos[t] + E^2 Cos[2 t] - 3 (2 + E^2 + 2 E t Sin[t]))
z[t] -> 1/144 a^3 ( ...)


But MMA can' t solve it.

• It's of a type that DSolve for which it is not programmed to solve. You might report it to WRI. It's perhaps a rare type, but one for which there is a programmatic approach, and maybe they haven't been motivated to implement a solution. One might call it a (lower) triangularizable system. – Michael E2 Jan 15 at 13:58

Clear[x, y, z];

• A more solve-like result: Fold[Join[#, DSolve[Sequence @@ (#2 /. First@#1), t], 2] &, {{}}, Thread[{{x''[t] + x[t] == a, y''[t] == x[t]^2 - y[t], z[t] == 2 x[t]*y[t] - z''[t]}, {x, y, z}}]] – Michael E2 Jan 15 at 13:55