Differential equation how to find out bounded or not bounded

Question

For which value of the parameter $k$ will all solutions of remain bounded as $t \rightarrow \infty$? $$u''(t)+k~u(t)=2 \sin(10t)?$$

Answer 1

Hints:

• Solve for the complementary solution.
• Solve for the particular solution.
• Form $y(t)$ and then analyze the limit against values of $k$.

The solution you should arrive at is:

$$y(t) = c_1 \cos(\sqrt{k} ~t) + c_2 \sin(\sqrt{k}~ t) + \dfrac{2}{k-100} \sin(10t)$$

Now, can you analyze what happens as $t \rightarrow \infty$ for different values of $k$ and figure where it remains bounded?