Linear differential equation

In mathematics, a linear differential equation is a differential equation

Lf = g,

where the differential operator L is a linear operator. The condition on L rules out operations such as taking the square of the derivative of f; but permits, for example, taking the second derivative of f. Therefore a fairly general form of such an equation would be

<math>D^n y(x) + a_{n-1}D^{n-1} y(x) + \ldots + a_1 D y(x) + a_0 D^0 y =g(x)<math>

where D is the differential operator d/dx, and the a_i are given functions. Such an equation is said to have order n, the index of the highest derivative of f that is involved.

The case where g = 0 is called a homogeneous equation, and is particularly important to the solution of the general case (by a method traditionally called particular integral and complementary function. When the a_i are numbers, the equation is said to have constant coefficients.

Homogeneous linear differential equation with constant coefficients

To solve such an equation one makes a substitution

y=e^λx,

to form the characteristic equation

<math>\lambda^n +a_{n-1}\lambda^{n-1}+\ldots+a_1\lambda+a_0<math>

to obtain the solutions

<math>\lambda=s_0, s_1, \ldots, s_{n-1}.<math>

When this polynomial has distinct roots, we have immediately n solutions to the differential equation in the form

<math>y_i(x)=e^{s_i x}.<math>

Then the general solution to the homogeneous equation can be formed from a linear combination of the y_i, ie.,

<math>y_H(x)=A_0 y_0(x)+A_1 y_1+\ldots+A_{n-1} y_{n-1}<math>

Where the solutions are not distinct, it may be necessary to multiply them by some power of x to obtain linear independence; the general solution therefore involves the product of polynomials and exponentials.

Inhomogeneous linear differential equation with constant coefficients

To obtain the solution to the inhomogeneous equation, find a particular solution y_P(x) by the product rule to

<math> D (y(x)e^{\int f(x)\,dx})=g(x)e^{\int f(x)\,dx}<math>

on integrating both sides yields

<math> y(x)e^{\int f(x)\,dx}=\int g(x)e^{\int f(x)\,dx} \,dx+c<math>

<math> y(x) = {\int ge^{\int f(x)\,dx} \,dx+c \over e^{\int f(x)\,dx}}<math>

See also:

• Laplace transform