MHB How to Solve the Differential Equation x'=x+sin(t)?

[karush]

Solve the differential equation
$x'=x+\sin(t)$
ok this uses x rather than y which threw me off
so rewrite as
$x'-x=\sin(t)$
thus $u(t)=e^{-t}$
$e^{-t}x'-e^{-t}x=e^{-t}\sin{t}$
and
$(e^{-t}x)'=e^{-t}\sin{t}$
intergrate thru
$\displaystyle e^{-t}x=\int {e^{-t} \sin(t)} dt = -1/2 e^{-t}\sin(t) - 1/2 e^{-t} \cos(t) +e^{-t} c$
divide thru
$\displaystyle x(t)=-\frac{\sin t}{2}-\frac{\sin t}{2}+ce^t$ok well typos probably

W|A returned $\displaystyle x(t) = c_1 e^t - \frac{\sin(t)}{2} - \frac{cos(t)}{2}$

 

[MarkFL]

Let's back up to:

$$\frac{d}{dt}\left(e^{-t}x\right)=e^{-t}\sin(t)$$

Now, when we integrate, we get:

$$e^{-t}x=-\frac{1}{2}e^{-t}\left(\sin(t)+\cos(t)\right)+c_1$$

Hence:

$$x(t)=c_1e^t-\frac{1}{2}\left(\sin(t)+\cos(t)\right)$$

This is equivalent to W|A returned. :)

 

[karush]

Isn't the 1/2 distributed?

 

[MarkFL]

Isn't the 1/2 distributed?

You could if you want. I chose not to.