How to find the closed form solution of the natural and forced response of 2sin(3t) - 4x with x(0) = 0

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So far I have used dsolve and have the following code:
syms x(t)
eqn = diff(x,t) == 2*sin(3*t)-4*x;
S = dsolve(eqn)
with the result as
S =
C1*exp(-4*t) - (2*cos(3*t + atan(4/3)))/5
However seeing the result, I am not very sure if I am doing this correctly. Any help would be greatly appreciated. Thanks!

Accepted Answer

John D'Errico
John D'Errico on 3 Sep 2021
Edited: John D'Errico on 3 Sep 2021
Why do you think it is not correct? Would dsolve lie? That dsolve code must be pretty darn tricky.
In fact, you have not used dsolve completely properly, because you never supplied the initial condition. You could have done so after the fact though, by solving for C in your solution such that the initial condition applies.
syms x(t)
eqn = diff(x) == 2*sin(3*t) - 4*x;
xsol = dsolve(eqn,x(0) == 0)
xsol = 
Is that the solution? Does it satisfy the initial condition?
subs(xsol,t,0)
ans = 
0
Clearly it satisfies the initial condition.
simplify(subs(eqn,x,xsol))
ans(t) = 
symtrue
It does satisfy the ODE.

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