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Home Differential Equations First Order Linear Equations Solving A First Order Linear Differential Equation
 
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Solving A First Order Linear Differential Equation

(1)

y'+ p(t)y = f(t).

The corresponding homogeneous linear differential equation is

(2)

x' + p(t)x = 0.

Step 1

Find a nonzero particular solution x(t) of the corresponding homogeneous linear differential equation (2). By the method of Section 14.2, we may take

x(t) = e-∫ p(t) dt.

Step 2

Find a function v(t) whose derivative is given by

14_differential_equations-59.gif

This is done by integration,

14_differential_equations-60.gif

Step 3

The general solution of (1) is y(t) = v(t)x(t) + Cx(t).

Step 4

If an initial value is given, the particular solution for the initial value problem is found by substituting and solving for the constant C.

Discussion Step 2 gives us a function v(t) for which v'(t)x(t) = f(t). Therefore, by

our previous discussion, v(t)x(t) is a particular solution of the linear equation (1).

Step 3 is then justified by Theorem 1.
Example 1
Example 2: Population Growth
Example 3

Home Differential Equations First Order Linear Equations Solving A First Order Linear Differential Equation

Last Update: 2006-11-17