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Suppose that b(t) is continuous and $\lim_{t \to \infty} b(t)=0$ and $\int_{0}^{\infty} |b'(t)| < \infty $ prove that every solution of the differential equation y"+(1+b(t))y=0 is bounded on $[0,\infty)$.

So started in this way: converting the above equation into

$A=\begin {bmatrix} 0 &1 \\ 1+b(t) & 0 \end {bmatrix} , x'=Ax $

which $x= (x_1,x_2)$ and obviously $x_1'=x_2$ and $x_1=y'$. And then I tried to use some theorems due to bounding the solutions of linear differential equations of order 1, But I have not any progress yet.

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    These kind of problems are usually solved using [Barbalat's lemma](https://en.wikipedia.org/wiki/Lyapunov_stability#Barbalat.27s_lemma_and_stability_of_time-varying_systems) – 2017-01-08
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    @polfosol How Barbalat's Lemma helps on this problem? Would you mind tell me more about your idea? – 2017-01-08
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    Posting an answer is really hard when I'm using the mobile app! See the example at the bottom of wikipedia article. Hopefully you'll get the idea – 2017-01-08
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    @polfosol yeah It's OK. But I'm waiting for your Idea. Pleas post it later If you could. – 2017-01-08

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