Solving differential equation using Laplace transform

Question

Can this DE be solved using Laplace transform?

$\frac{\mathrm{d} y}{\mathrm{d} x}\cos x=y\sin x+\cos ^{2}x$

Answer 1

In the form it is in right now, I do not think the Laplace transform is of use. However, you can rewrite it in such a way to use the Laplace transform:

$$-y\sin x + y'\cos x = \cos^2 x \Longrightarrow \frac{d}{dx}(y\cos x) = \cos^2 x.$$

From here you can use a Laplace transform. You might want to use a double angle identity to simplify $\cos^2 x$.

That said, unless you are specifically instructed to use the Laplace transform, I do not advise using it. It's like killing a mosquito with a rocket launcher when you could just use bug spray. You're using a lot of machinery to do something that is done so much simpler with easier techniques.