﻿ Variation of parameters

# Variation of parameters

The variation of constants is a method from the theory of linear ordinary differential equations for determining a particular solution of an inhomogeneous linear system of differential equations of first order and an inhomogeneous linear differential equation of arbitrary order. Assuming this is a complete solution ( fundamental system ) of the associated homogeneous differential equation.

The procedure was developed by the mathematician Joseph -Louis Lagrange.

• 2.1 formulation
• 2.2 proof
• 2.3 Special case: the case of resonance
• 3.1 formulation
• 3.2 proof
• 3.3 Alternative: basic solution method
• 3.4 proof

## Motivation

### Linear first order differential equation

Consider the scalar linear differential equation of first order

Next is a primitive function of, for example,

Then

The set of all solutions of the homogeneous differential equation. As an approach to the solution of the inhomogeneous problem set you

That is, you can vary the constant. This provides a unique mapping between the functions and, as is always a positive, continuously differentiable function. The derivation of this trial function

So solve the inhomogeneous differential equation

If and only if

Applies. For example, is

Such a function, and thus

The special solution. So is

The set of all solutions of the inhomogeneous differential equation.

### Example

Is situated on a coil to the inductor and the electric resistance to a DC voltage, then for the voltage across the coil

According to Ohm's law therefore applies

It is an ordinary, inhomogeneous, linear first order differential equation with constant coefficients, which will now be solved using the method of variation of constants.

The associated homogeneous equation is

It follows that

For each constant is a solution of the homogeneous problem.

As an approach to the solution of the inhomogeneous equation we replace the constant by a variable expression. So you set

And attempts to determine a differentiable function so that the inhomogeneous differential equation is satisfied. It follows

So the differential equation is exactly satisfied when

Is so synonymous with, from which we obtain by integration. Thus, the inhomogeneous differential equation is

Solved. The constant can not be determined from initial conditions. For example, results for the solution

## Inhomogeneous linear systems of differential equations of first order

The above method can be generalized in the following way:

### Formulation

Let and be continuous functions and a fundamental matrix of the homogeneous problem and that matrix obtained from by replacing the th column of. Then

With

The solution of the inhomogeneous initial value problem.

### Evidence

Set

It is, and because we see by differentiating that the differential equation is satisfied. Now solves

For solid system of linear equations

According to Cramer's rule is thus

So true

### Special case: the case of resonance

If the inhomogeneity is itself solution of the homogeneous problem, ie, this is referred to as resonance. In this case,

The solution of the inhomogeneous initial value problem.

## Inhomogeneous linear differential equations of higher order

Solving a differential equation of higher order is equivalent to solving an appropriate system of differential equations of first order. In this way, one can use the above method to construct a special solution for a differential equation of higher order.

### Formulation

Be continuous functions and a fundamental matrix of the homogeneous problem, whose first line reads, as well as that matrix obtained from by replacing the th column of. Then

With

The solution of the inhomogeneous initial value problem.

### Evidence

Consider first the differential equation corresponding to this first-order system consisting of equations

The following applies: solves the scalar equation of order if and only if solution is above system of first order. By definition, is a fundamental matrix for this system of first order. Then finally we turn to the above- proven method of variation of constants.

### Alternative: basic solution method

In the case of constant coefficients, it is sometimes advantageous to use the fundamental solution method for constructing a particular solution: Is that a homogeneous solution of which

Fulfilled, then

That particular solution of with.

### Evidence

By differentiating checked one

And

The result is

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