5. Homogeneous equation

Homogeneous equations can be written in the form: , and also in the form

М(x, y)dx+N(x, y)dy=0,

where М(x, y) and N(x, y) - homogeneous functions of the same degree.

To solve the homogeneous equation, we must make the substitution y = tx, then we obtain an equation with separable variables. At this change dy=tdx+xdt.

Problem Statement: Given a first order linear differential equation and (possibly) an initial condition of the form

, .

Find the general solution for the given equation. That is, find the one-parameter family of functions that are solutions of the given equation.

Notice that the terms involving and y are on the left-hand side of the equation and the only variable on the right-hand side is the independent variable t.

Step 0: Put the problem into standard form.

If the problem is presented in any form other than the one described above, re-write the problem in the form above first. Then divide the equation through by making the coefficient of exactly one. This means that the problem has the (standard) form

, .

Step 1: Determine the integrating factor, i.e., .

You will need to evaluate the integral by techniques from the calculus.

Step 2: Multiply the equation through by the integrating factor found in the previous step.

This step is straight forward algebra. Just remember to multiply both sides of the equation by .

Step 3: Integrate both sides of the equation.

You should check (by mentally differentiating) that the left hand side is exactly the derivative of the product . If it isn’t, you have made a mistake in Steps 1 – 3. Don’t forget the constant of integration.

Step 4: Solve (algebraically) the result in Step 3 for y explicitly.

The result is the general solution for the equation.

Step 5: (only if an IVP is given) Apply the initial conditions and solve (algebraically) for the value of the constant of integration.

This is accomplished by substituting for t and for y. The constant of integration is the unknown in the resulting equation.

A second order linear differential equation with constant coefficients is a differential equation of the form

,

where and are constants.

The characteristic equation associated with the differential equation above is

.

This quadratic has two (real or imaginary) roots: and .

If , then the general solution to the second order differential equation above is of the form

.

If , then the general solution is of the form

.

Example 9. Solve the following initial value problem:

The characteristic equation is

,

so that the characteristic equation has two roots: and

The general solution to the second order linear differential equation is therefore

In order to solve the initial value problem, we need to find the constants and :

The solution to the initial value problem is therefore

6. Non-homogeneous second-order equations

Special case of the general linear equation of order n

(1)

where and are given functions of or are constants.

Consider equation (1). If , it is called homogeneous equation with variable or constant coefficients, depending on whether are functions of or are constants. It is also called the reduced linear differential equation.

Example 10. ,is a second order homogeneous equation with variable coefficient.

If in equation (1), it is called non- homogeneous equation with variable or constant coefficients, depending on whether are functions of or are constants. Example 11. is a second order non-homogeneous equation with variable coefficient.

Suppose and are two independent solutions of the reduced equation of (1), namely

(2)

then the linear combination where are arbitrary constants, is also a solution.

Theorem 1. If are n linearly independent functions of which satisfy a homogeneous equation (3), then the linear combination

(3)

where are arbitrary constants, is its solution. Equation (3), which provides the solution for the homogeneous equation is called the complementary function.

Theorem 2. The general solution of a complete non-homogeneous differential equation is equal to the sum of its complementary function and any particular integral. If is a particular solution of (1), then the general solution is

+ (4)

Hence, for non-homogeneous equations:

General solution = Complementary function + Particular integral

The general linear equation (4) is usually difficult to solve and requires special techniques. However an important and special case occurs when the coefficients are constants, the equation being called a constant coefficient equation. Consider a constant coefficient homogeneous equation

(5)

Denoting, , in equation (5) we have

. (6)

If we make a formal substitution in (6), we obtain a polynomial in

of degree given:

, (7)

and if we equate equation (7) to zero we have an algebraic equation of degree which must have roots. The equation is called the auxiliary equation of the differential equation of the differential equation (7).

Theorem 3. If is a root of the auxiliary equation , then is a solution of the homogeneous linear differential equation

where is a constant.

Summary to solving a homogeneous differential equation

The solution of the differential equation then reduces to solving the algebraic auxiliary equation for its roots and forming a linearly independent combination as the general solution, if the roots were all distinct.

Theorem 4. If the auxiliary equation of a homogeneous linear differential equation contains as an s-fold root, then is a solution of the differential equation.

Example 12. If the (twice), the solution is

Auxiliary equation with complex roots

If the auxiliary equation with real coefficients contains two complex roots and then is a solution of the differential equation, are constants.

Generalization

If the complex roots occur as multiple roots, that is if ( is an s-fold pair of complex roots, then the corresponding terms of the complementary function are

Equations with undetermined coefficients

In the previous section we learnt that solution to the complete linear differential equation is composed of the sum of the complementary function and the particular integral. Techniques for obtaining the complementary function were developed in sections 2.1.4-2.1.6 with many working examples. What remains is only to provide techniques for finding a particular integral in order to obtain a complete solution. In this section we shall discuss the technique called the method of undetermined coefficients.

Although the method of undetermined coefficients is not applicable in all cases, it may be used if the right- hand side ,contains only terms which have a finite number of linearly independent derivatives such as , , , or products of these.

Procedure for the techniques of undetermined coefficients

The general procedure in this technique is to assume the particular integral of a form similar to the right member in equation (1). The necessary derivatives of are then obtained and substituted into the given differential equation. This results in an identity for the independent variable, and consequently the coefficients of the like terms are equated. The values of the undetermined coefficients are found from the resulting system of linear equations. The procedure is best illustrated with an example

Table 1 below summarizes a general rule for the formulation of the particular integral

If is of the form	Choose to be

Table 1

Example 13: Find the particular integral of

Solution: You actually worked out the complementary function of this equation to be .That is, the auxiliary equation is , giving or .Hence the complementary function is as before. Looking at the right hand side of the above differential equation example 13.3.1 and the general rule in table 1, the particular integral is

, ,

Substituting in the original equation

Comparing the coefficients on both sides

The general solution is then the complementary function + particular integral, which is

.

_{LECTURE 12.}

Numerical series. The convergence of the sum of the series. Alternating series. Absolute and conditional convergence of the series. Functional series. The area of series convergence. Power series. The radius of convergence