Initial-value problem

In the context of ordinary differential equations, we sometimes have additional conditions that our unknown function $y(x)$ or its derivatives must satisfy, called initial-value problems.

Because we have general solutions for ODEs, we often express in terms of an arbitrary constant. To find the value of an arbitrary constant, we are given an initial value that constrains the solution to one function. For any $n$-th order ODE, we’ll have $n$ initial values.