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Home Differential Equations First Order Linear Equations Examples Example 2 - Population Growth
 
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Example 2 - Population Growth

A population has a net birthrate of 2% per year and a net annual immigration rate of 100,000 sin t. At time t = 0 years, the population is 1,000,000. Find the population as a function of t.

The given verbal problem can be expressed as the initial value problem y' = 0.02y + 100,000 sin t, y(0) = 1,000,000.

We first put the equation in the usual form with all the y terms on the left side,

y' - 0.02y = 100,000 sin t, y(0) = 1,000,000.

Step 1

The corresponding homogeneous equation is x' - 0.02x = 0. The particular solution is x(t) = e0.02t.

Step 2

14_differential_equations-64.gif

v can now be found by integration by parts. With u = sin t and dw = e-0.02t dt, we have w = -50e-0.02t and

14_differential_equations-65.gif

Similarly,

14_differential_equations-66.gif

Combining the last two equations and solving for the integral of sin te-0.02t, we get

14_differential_equations-67.gif

Step 3

The general solution is y = vx + Cx, or

14_differential_equations-68.gif

Step 4

Substitute at t = 0.

14_differential_equations-69.gif

The particular solution (Figure 14.3.2) is then

14_differential_equations-70.gif

14_differential_equations-71.gif

Figure 14.3.2 Example 2
Home Differential Equations First Order Linear Equations Examples Example 2 - Population Growth

Last Update: 2006-11-17