# 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

[itex]D^n y(x) + a_{n-1}(x)D^{n-1} y(x) + \cdots + a_1(x) D y(x) + a_0(x) D^0 y =g(x)[itex]

where D is the differential operator d/dx, and the ai 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 ai are numbers, the equation is said to have constant coefficients.

 Contents

## Homogeneous linear differential equation with constant coefficients

To solve such an equation one makes a substitution

y = eλx,

to form the characteristic equation

[itex]\lambda^n +a_{n-1}\lambda^{n-1}+\cdots+a_1\lambda+a_0[itex]

to obtain the solutions

[itex]\lambda=s_0, s_1, \dots, s_{n-1}.[itex]

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

[itex]y_i(x)=e^{s_i x}.[itex]

It is easy to see that these are then linearly independent, by applying the Vandermonde determinant. Therefore their linear combinations, with n coefficients, should provide a complete solution. So it proves: it is known that the general solution to the homogeneous equation can be formed from a linear combination of the yi, ie.,

[itex]y_H(x)=A_0 y_0(x)+A_1 y_1+\cdots+A_{n-1} y_{n-1}[itex]

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, of degrees bounded in terms of the multiplicities of the roots, and exponentials.

## Inhomogeneous linear differential equation with constant coefficients

To obtain the solution to the inhomogeneous equation, find a particular solution yP(x) by the method of undetermined coefficients; the general solution to the linear differential equation is the sum of the homogeneous and the particular solution.

## Other meanings

The term linear differential equation can also refer to an equation in the form

[itex] Dy(x) + f(x) y(x) = g(x)[itex]

where this equation can be solved by forming the integrating factor

[itex]e^{\int f(x)\,dx}[itex],

multiplying throughout to obtain

[itex] Dy(x)e^{\int f(x)\,dx}+f(x)y(x)e^{\int f(x)\,dx}=g(x)e^{\int f(x)\,dx}[itex]

which simplifies due to the product rule to

[itex] D (y(x)e^{\int f(x)\,dx})=g(x)e^{\int f(x)\,dx}[itex]

on integrating both sides yields

[itex] y(x)e^{\int f(x)\,dx}=\int g(x)e^{\int f(x)\,dx} \,dx+c[itex]
[itex] y(x) = {\int ge^{\int f(x)\,dx} \,dx+c \over e^{\int f(x)\,dx}}[itex]

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