Initial Conditions

Setting the stage for unique solutions

Within some systems there will be further limitations on on how solutions to a Differential Equation can be defined. Many systems require, or naturally impose, initial conditions.

Ordinary Differential Equations

In ODEs, initial conditions are used to specify the values of the dependent variable(s) at a specific time, often denoted as

` t_0 `

This sets the starting point for the equation's solution. For example, in a first-order ODE such as

` \frac{dy}{dt} = f(t, y) `

an initial condition might be given as

` y(t_0) = y_0 `

This states that at time t_0 the value of y is y_0.

Example

For the equation

` \frac{dy}{dt} = 3y `

with the initial condition

` y(0) = 4 `

the solution reflects that at time t=0, y "starts" at 4.

Partial Differential Equations

In PDEs, initial conditions can be more complex due to the involvement of multiple independent variables. For time-dependent PDEs, the initial condition often specifies the state of the function across all spatial variables at the initial time.

Example 1

For the heat equation

` \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} `

an initial condition might be

` u(x, 0) = f(x) `

indicating the temperature distribution along a rod at t=0.

NOTE: the initial conditions are fixed in time, but the initial heat distribution still depends on a displacement from a specific point, x.

Example 2

In a wave equation such as:

` \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} `

initial conditions could include both the initial displacement:

` u(x, 0) = g(x) `

and the initial velocity:

` \frac{\partial u}{\partial t}(x, 0) = h(x) `

In both ODEs and PDEs, these initial conditions are sometimes essential for the uniqueness and existence of a solution. They can, in many applied situations, allow experiments to taylor general solutions to specific situations.