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Solving First Order Homogeneous Linear Differential Equation

Method For Solving First Order Homogeneous Linear Differential Equation

(1)

The general solution is

y(t) = Ce-Pω,

where P(t) is an antiderivative of p(t). That is,

y(t) = Ce-∫p(t) dt.

This formula is obtained by the procedure described in Section 14.1 for differential equations with separable variables, as follows. First write the equation in the form

Step 1

There is a constant solution y(t) = 0.

Step 2

Separate the variables and integrate: y-1 dy= -p(t)dt.

ln|y|= -∫ p(t) dt + B.

Now solve for y.

|y| = e-∫ p(t) dt + B,

y = Ce-∫ p(t) dt,

where

C = eB if y > 0, and C = -eB if y < 0.

Step 3

Combining Steps 1 and 2, we get the general solution

y(t) = Ce-∫ p(t) dt.
Remark

The case C = 0 gives the constant solution y(t) = 0 of Step 1. Discussion The constant of integration in the indefinite integral

∫ p(t) dt

will be absorbed in the constant C.

The particular solution for the initial value y(t0) = y0 is found by substituting and computing C. Notice that any two particular solutions of the same homogeneous linear differential equation differ only by a constant factor. If x(t) is any nonzero particular solution, then the general solution is Cx(t).
Home Differential Equations First Order Homogenous Linear Equations Solving First Order Homogeneous Linear Differential Equation

Last Update: 2006-11-17