Applying initial conditions to differential equations

Thus far, we have stopped solving a differential equation once we have identified the family of solutions which satisfy the original equation. However, we have not yet explored how we can narrow down our solution to a limited number of these. To do this, we need to be given some more information about the solution, for example the value of $y_0 = y(x_0)$ for a particular $x_0$ . For example, we might be told that

\frac{dy}{dx} = y, \quad y(1) = 3

This information, if we are given it, is called an initial condition. The name is a bit odd, but it comes from the application of differential equations to answer questions about systems that change over time. It is often the case that the variable $x$ represents time, and so if we know the state of the system before it starts to change, we can solve the differential equation to determine how this state will change over time.

In practice, the initial condition is usually used at the end of a solution to determine the value of the constants of integration, which is why we have been paying careful attention to which values of these are valid in our previous examples.

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