Reduction of order

The reduction method of d' Alembert, a method from the theory of ordinary differential equations, which is named after the mathematician and physicist Jean -Baptiste le Rond d' Alembert. It is used to return a linear differential equation of order with non- constant coefficients with knowledge of a solution of the homogeneous problem on a linear differential equation of order.

Roughly speaking, the following applies: To a ( inhomogeneous ) to solve linear differential equation of order, supply them to a non-trivial solution of the associated homogeneous linear differential equation. Then the variation of parameters approach to the original equation leads to a ( non-homogeneous ) linear differential equation of lower order for.

Wording of the sentence

Consider the differential operator of order

For this is a solution of the homogeneous linear differential equation

Known. for

Then applies

In other words, solve the inhomogeneous differential equation of order if and only if

The inhomogeneous linear differential equation of order

Solves.

Evidence

After Leibniz's rule

So

Now, by assumption. It thus follows

Index shifting supplies.

Special case: Linear second order differential equation

Be solution of the homogeneous linear differential equation of second order

Then

Solution of the ( inhomogeneous ) differential equation

If and only if

The equation

Sufficient. This equation can be completely solved by the variation of constants.

Generalizations

There is also a generalization for linear systems of equations.