In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x.
Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives.
Consider the differential equation ## \ddot{x}+cos(x)=0 ##.
Note that ## \ddot{x}=f(x, \dot{x}) ##, so we have ## f(x, y)=-cos(x) ##.
Then ## f(x, 0)=-cos(x)=0 ##.
This gives ## x=n\pi-\frac{\pi}{2} ## for some ## n\in\mathbb{Z} ##.
Since the differential equation for the phase paths is given by...