# Why the result by hand is different from the command “DSolve”?

Yesterday,I use the Mathematica to solve a Differential Equation by the command DSolve:

DSolve[y''[x] + 2 y'[x] + 5 y[x] == x Cos[x], y[x], x]


{{y[x] -> E^-x C[2] Cos[2 x] + E^-x C1 Sin[2 x] + 1/200 (-25 Cos[x] Cos[2 x] + 25 x Cos[x] Cos[2 x] - 3 Cos[2 x] Cos[3 x] + 15 x Cos[2 x] Cos[3 x] - 25 x Cos[2 x] Sin[x] + 25 x Cos[x] Sin[2 x] + 4 Cos[3 x] Sin[2 x] + 5 x Cos[3 x] Sin[2 x] - 25 Sin[x] Sin[2 x] + 25 x Sin[x] Sin[2 x] - 4 Cos[2 x] Sin[3 x] - 5 x Cos[2 x] Sin[3 x] - 3 Sin[2 x] Sin[3 x] + 15 x Sin[2 x] Sin[3 x])}}

However,I feel the result is very complex.So I caculate it by hand.Below is my process.

Characteristic equation: r^2+2*r+5==0 r1=-1+2 I,r2=-1-2 I

So the Nonhomogenerous equation general solution is: E^-x(C1 Cos[2x]+C2 Sin[2x])

I use the Differential Operator method to caculate the nonhomogenerous particular solution:

So my question is why the two result is different?Can someone tell me the reason? Thanks sincerely!!

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If you simplify M answer, then it becomes: Simplify[%]; y[x] -> 1/50 E^-x (50 C[2] Cos[2 x] + E^x (-1 + 5 x) Sin[x] +Cos[x] (E^x (-7 + 10 x) + 100 C[1] Sin[x])) –  Nasser Nov 17 '13 at 3:35

They are the same answer, as verified below. You can use the Simplify command to simplify things if needed:

mma = y[x] /. First@DSolve[y''[x] + 2 y'[x] + 5 y[x] == x Cos[x], y[x], x];
handH = E^(-x)*(C[2] Cos[2 x] + C[1] Sin[2 x]);
handP = (1/5 x - 7/50) Cos[x] + (1/10 x - 1/50) Sin[x];
hand = handH + handP;
Simplify[mma - hand]


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