Let $f(t) = 2\pi \sin t$, and determine a $2\pi$-periodic function $y^∗$ with the property that $\lim_{t\to+\infty} |y(t) − y^∗(t)| = 0$ for every solution $y$ of $y′ + y = f$.
I am having trouble with this question. I don't even know how to start it. Any help or hints would be greatly appreciated.
This is a linear first order differential equation.
The general solution is $c\cdot y_h(t) + y_p(t)$ where $y_h(t)$ solves the homogenous problem $y^\prime + y = 0$ and $y_p$ is a particular solution to your problem.
Since $y_h(t) = e^{-t}$ goes to $0$ as $t\to\infty$, every solution will tend to $y_p(t)$ as $t\to\infty$.
One way to find $y_p(t)$ is to guess a solution of the form $y_p(t) = A\cos(t) + B\sin(t)$, and use the differential equation to solve for $A$ and $B$.