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: Givena 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 theintegrating 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

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