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