Linear differential equation

From Academic Kids

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

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

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

to obtain the solutions

<math>\lambda=s_0, s_1, \dots, 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>

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.,

<math>y_H(x)=A_0 y_0(x)+A_1 y_1+\cdots+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, 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

<math> Dy(x) + f(x) y(x) = g(x)<math>

where this equation can be solved by forming the integrating factor

<math>e^{\int f(x)\,dx}<math>,

multiplying throughout to obtain

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

which simplifies due to 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

ja:線型微分方程式

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