Linearizaton of Initial Value Problems

If one does not require or cannot find the exact solution that gives an explicit formula for an initial value problem, numerical methods (approximation methods) can yield close results.

Linearization

Given a differential equation $y^{\prime }=f(x,y)$ and an initial condition $y(x\_0)=y\_0$ one can approximate the solution by its linearization (Tangent line to the solution)

$$L(x)=y(x\_0)+y^{\prime }(x\_0)(x-x\_0)$$

This gives a good approximation to the solution $y(x)$ in a short interval about $x\_0$