2
$\begingroup$

I need to solve a differential equation on the following form

$\frac{d^2 f(x)}{dx^2}=A\sin(f(x))+B\cos(f(x))$

Where $A$ and $B$ are constants. I am not sure how to approach this, which method to use etc. And I am not even sure if it is possible to solve analytically.

  • 0
    Robert Israel has given some of details. When you multiply both sides by $\frac{dy}{dx}$, the left side is the derivative of $(1/2)\left(\frac{dy}{dx}\right)^2$, the right is the derivative of $-A\cos y+B\sin y$. So we get $(1/2)\left(\frac{dy}{dx}\right)^2=-A\cos y+B\sin y +C$.2012-10-17

1 Answers 1

2

With $y = f(x)$ and $v = df/dx$, this autonomous second-order DE becomes $ \dfrac{dv}{dy} = \frac{A \sin(y) + B \cos(y)}{v}$ and that is separable, so $ \frac{v^2}{2} = \int (A \sin(y) + B \cos(y)) \ dy = - A \cos(y) + B \sin(y) + C$

  • 0
    @Robert: Yes, I am working with a pendulum in physics and arrived at this differential equation. Thanks for the help.2012-10-18