Initial Value Problems

With $n$ initial conditions for an $n^{th}$ order IVP, the family of Solutions is narrowed down to a particular solution

Form

Given the nth-order differential equation

$$\frac{d^{n}y}{d{x}^n}=f(x,y,y^{\prime },\cdots ,y^{(n-1)})$$

find a solution to the DE on an interval I that satisfies at $x_0$ the initial conditions:

$$\begin{array}{r}y(x_0)=y_0\\ y^{\prime }(x_0)=y_1\\ ⋮\\ y^{n-1}(x_0)=y_{n-1}\end{array}$$

where $x_0\in I$ and $y_0,y_0,\cdots ,y_n$ are given constants