Solving differential equations with trigonometric functions using laplace

Question

Can this DE be solved using Laplace transform?

$$ L\;\frac{d^2\theta}{dt^2} + A\cos(\theta) +g\sin(\theta) = 0 $$

where g , A , L are constants

Answer 1

No, the Laplace transform works for linear equations.

You can handle your equation as $$\dot\theta\ddot\theta+\dot\theta(a\cos\theta+b\sin\theta)=0$$ and by integration $$\dot\theta^2=2a\sin\theta-2b\cos\theta+c.$$

This separable equation leads you to an elliptic integral, then to a Jacobi function, something not too exotic. (Note that by adding a suitable constant to $\theta$, you can let one of the sine or cosine term vanish.)