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Example 3

Find the general solution of the equation y' - sy = Kert, where r, s, and K are constants.

Step 1

The corresponding homogeneous equation is x' - sx = 0. It has the particular solution x(t) = est.

Step 2

14_differential_equations-72.gif

There are two cases, r ≠ s and r = s.

Step 3

(Case 1) r ≠ s.

14_differential_equations-73.gif

The general solution y = vx + Cx in this case is

14_differential_equations-74.gif

Step 3

(Case 2) r = s. In this case v'(t) = K, and v(t) = Kt. The general solution in this case is

y(t) = Ktest + Cest.

We now return to the general first order linear differential equation (1). Using definite integrals, we can get a single formula for the solution of equation

(1) 

y' + p(0)y = f(t)

by combining Steps 1 to 4.

For Step 1, choose an initial point a, and get a particular solution of the corresponding homogeneous equation,

For Step 2, write v(t) as a definite integral from a to t,

14_differential_equations-75.gif

Step 3 shows that the general solution is y = vx + Cx, and the final formula is found by substituting for v and x.
Home Differential Equations First Order Linear Equations Examples Example 3

Last Update: 2006-11-17