# How can I get Dsolve's output to look like traditional solution?

I am still quite new to Mathematica, so please bear with me. I am using Dsolve on a 3D linear differential equation and got what appeared to be an incorrect answer. After some work, it seems that the solution is, in fact, correct, but is not in the typical form of exponentials and sine/cosine I expected from a system with distinct roots and imaginary eigenvalues. It is concerning to see a solution to a linear ODE (without repeated roots) that has $te^{\lambda t}$ and $t^2e^{\lambda t}$ in it. Is there a way to change the output of Dsolve to be more in line with what I expect?

Here is the code I use to produce the questionable output.

A = {{-10, 0, 75/23}, {2, -10, 77/92}, {0, 8, -1}}
SYST = A.{x[t], y[t], z[t]}
SOL = DSolve[{
x'[t] == SYST[[1]],
y'[t] == SYST[[2]],
z'[t] == SYST[[3]],
x[0] == x0, y[0] == y0, z[0] == z0
},
{x[t], y[t], z[t]}, t];

• Depending on what form you want, try SOL // N or ComplexExpand[SOL // N] or ComplexExpand[sol // N] // FullSimplify – bill s Nov 30 '17 at 21:12

One way to simplify it could be

ClearAll[x,t,y,z,z0,x0,y0]
A0    = {{-10,0,75/23},{2,-10,77/92},{0,8,-1}};
syst  = A0.{x[t],y[t],z[t]};
ode   = {x'[t]==syst[[1]],y'[t]==syst[[2]],z'[t]==syst[[3]]};
ic    = {x[0]==x0,y[0]==y0,z[0]==z0};
sol   = First@DSolve[{ode,ic},{x[t],y[t],z[t]},t]


And now

 Chop@FullSimplify@ComplexExpand@N@sol


• Perfect, thank you very much! – user53868 Dec 1 '17 at 14:12