# Initial value problem

## Contents

## Definition

An **initial value problem** in the context of a differential equation (here, an ordinary differential equation) is the following data:

- A differential equation UNIQ2fed201a894895b6-math-00000000-QINU (the independent variable here is UNIQ2fed201a894895b6-math-00000001-QINU and the dependent variable is UNIQ2fed201a894895b6-math-00000002-QINU). This is a differential equation of order UNIQ2fed201a894895b6-math-00000003-QINU.
- A tuple of numbers UNIQ2fed201a894895b6-math-00000004-QINU

### Solution concept

A solution to this is a functional (or relational) solution to the original differential equation such that at UNIQ2fed201a894895b6-math-00000005-QINU, we have UNIQ2fed201a894895b6-math-00000006-QINU and the UNIQ2fed201a894895b6-math-00000007-QINU derivative is UNIQ2fed201a894895b6-math-00000008-QINU for UNIQ2fed201a894895b6-math-00000009-QINU. More concretely, a function UNIQ2fed201a894895b6-math-0000000A-QINU solves the initial value problem if it solves the differential equation *and* UNIQ2fed201a894895b6-math-0000000B-QINU, i.e., UNIQ2fed201a894895b6-math-0000000C-QINU for UNIQ2fed201a894895b6-math-0000000D-QINU.

### Intuition behind going one derivative less

Note that the typical setup of an initial value problem specifies derivatives only up to the UNIQ2fed201a894895b6-math-0000000E-QINU for an order UNIQ2fed201a894895b6-math-0000000F-QINU differential equation. The intuition is that the differential equation controls the UNIQ2fed201a894895b6-math-00000010-QINU (and higher) derivatives in terms of the derivatives up to the UNIQ2fed201a894895b6-math-00000011-QINU. In the case of an explicit differential equation, the UNIQ2fed201a894895b6-math-00000012-QINU and higher derivatives are uniquely determined. In general, they are usually determined up to a finite or discrete ambiguity.

## Solution strategy

### In terms of the general solution

The typical solution strategy for an initial value problem is as follows:

- First, find the general solution. For a differential equation of order UNIQ2fed201a894895b6-math-00000013-QINU, we typically expect this to involve UNIQ2fed201a894895b6-math-00000014-QINU free parameters.
- Plug in the conditions UNIQ2fed201a894895b6-math-00000015-QINU. These are UNIQ2fed201a894895b6-math-00000016-QINU conditions. Each condition gives an equation involving the UNIQ2fed201a894895b6-math-00000017-QINU free parameters. This gives a system of UNIQ2fed201a894895b6-math-00000018-QINU equations in UNIQ2fed201a894895b6-math-00000019-QINU variables.
- Solve the system. Assuming consistent and irredundant equations, we expect the solution set to be zero-dimensionsal, which typically means it is finite or discrete. In good situations, we may get a unique solution.
- Having found the solutions for the free parameters, plug these in to get the functional (or relational) solutions to the initial value problem.

### Other strategies

There are some strategies to solve initial value problems that do not rely on finding the general solution. Generally, these involve doing definite instead of indefinite integrations in the process of finding the solution. This way we avoid introducing unnecessary constants. In some situations, this can save us from the effort of making cases based on the signs of those unknown constants, because definite integration avoids the introduction of unnecessary parameters.

## Variations

Instead of a single initial value problem, we may be given the values of the function and some of its derivatives at *multiple* points. The trade-off is that fewer derivatives may be specified. The solution method is the same: find the general solution, then plug in the conditions to get equations for the free parameters. In general, we would like to have UNIQ2fed201a894895b6-math-0000001A-QINU conditions in total (values of functions and derivatives) in order to get a unique, or finite, set of possible solutions.

One extreme case of a multiple initial value problem is the case where the value of the function (but no derivatives) is specified at UNIQ2fed201a894895b6-math-0000001B-QINU distinct points of the domain.

## Examples

### Example of a separable differential equation

This is a first-order differential equation, so there will be one free parameter, and the initial value specification only involves specifying the function value (no derivatives) at one point.

Consider the separable differential equation:

UNIQ2fed201a894895b6-math-0000001C-QINU

This has a stationary solution UNIQ2fed201a894895b6-math-0000001D-QINU. The relational solution family is given by:

UNIQ2fed201a894895b6-math-0000001E-QINU

Integrating:

UNIQ2fed201a894895b6-math-0000001F-QINU

Exponentiate:

UNIQ2fed201a894895b6-math-00000020-QINU

Let UNIQ2fed201a894895b6-math-00000021-QINU and get (**NOTE**: This UNIQ2fed201a894895b6-math-00000022-QINU has nothing to do with the UNIQ2fed201a894895b6-math-00000023-QINU denoting the order of a differential equation, used elsewhere on this page):

UNIQ2fed201a894895b6-math-00000024-QINU

So:

UNIQ2fed201a894895b6-math-00000025-QINU

We can combine back the stationary solution by allowing UNIQ2fed201a894895b6-math-00000026-QINU to get:

UNIQ2fed201a894895b6-math-00000027-QINU

We can see from this that every initial value specification yields a unique value of UNIQ2fed201a894895b6-math-00000028-QINU (we essentially get a linear equation in UNIQ2fed201a894895b6-math-00000029-QINU), and hence every initial value problem for this differential equation has a unique solution.

Consider two different initial value problems associated with this differential equation:

- UNIQ2fed201a894895b6-math-0000002A-QINU: In this case, we get:

UNIQ2fed201a894895b6-math-0000002B-QINU

Plug back into the original to get the unique solution to the initial value problem:

UNIQ2fed201a894895b6-math-0000002C-QINU

We can verify that this is both a solution to the differential equation and satisfies the initial value condition.

- UNIQ2fed201a894895b6-math-0000002D-QINU: In this case, we get:

UNIQ2fed201a894895b6-math-0000002E-QINU

So we get the stationary solution UNIQ2fed201a894895b6-math-0000002F-QINU.

This is as expected -- for any initial value condition where the initial value of UNIQ2fed201a894895b6-math-00000030-QINU is a stationary solution constant, the stationary solution should be one solution to the initial value problem. In our case, every initial value problem has a unique solution, so it is the only solution.