**Problem** We wish to solve the first-order [[Initial Value Problem|IVP]] of the form $\dot{y}= \frac{dy}{dt} = f(y,t)$ where $y=(y_{1}(t),y_{2}(t),\dots,y_{n}(t)) \in \mathbb{R}^{n}$, $f: [t_{0},\infty]\times \mathbb{R}^{n} \to \mathbb{R}^{n}$ and the initial conditions $y_{0} \in \mathbb{R}^{n}$. Without loss of generality to higher-order systems, we restrict ourselves to _first-order_ differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables. For example, the second-order equation $y'' = -y$ can be rewritten as two first-order equations: $y'=z$ and $z'=-y$. This process is known as [[Reduction of Order of Scalar Ordinary Differential Equations]]. ### Methods - [[Forward Euler Method]]. - [[Backward Euler Method]]. - [[Perturbed ODE]].