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Mathematics LibreTexts

17.2: First Order Homogeneous Linear Equations

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A simple, but important and useful, type of separable equation is the first order homogeneous linear equation:

Definition: first order homogeneous linear differential equation

A first order homogeneous linear differential equation is one of the form

˙y+p(t)y=0

or equivalently

˙y=p(t)y.

"Linear'' in this definition indicates that both ˙y and y occur to the first power; "homogeneous'' refers to the zero on the right hand side of the first form of the equation.

Example 17.2.2

The equation ˙y=2t(25y) can be written ˙y+2ty=50t. This is linear, but not homogeneous. The equation ˙y=ky, or ˙yky=0 is linear and homogeneous, with a particularly simple p(t)=k.

Because first order homogeneous linear equations are separable, we can solve them in the usual way:

˙y=p(t)y1ydy=p(t)dtln|y|=P(t)+Cy=±eP(t)y=AeP(t),

where P(t) is an anti-derivative of p(t). As in previous examples, if we allow A=0 we get the constant solution y=0.

Example 17.2.3

Solve the initial value problems ˙y+ycost=0, y(0)=1/2 and y(2)=1/2.

Solution

We start with

P(t)=costdt=sint,

so the general solution to the differential equation is

y=Aesint.

To compute A we substitute:

12=Aesin0=A,

so the solutions is

y=12esint.

For the second problem,

12=Aesin2A=12esin2

so the solution is

y=12esin2esint.

Example 17.2.4

Solve the initial value problem y˙y+3y=0, y(1)=2, assuming t>0.

Solution

We write the equation in standard form: ˙y+3y/t=0. Then

P(t)=3tdt=3lnt

and

y=Ae3lnt=At3.

Substituting to find A: 2=A(1)3=A, so the solution is y=2t3.

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This page titled 17.2: First Order Homogeneous Linear Equations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David Guichard via source content that was edited to the style and standards of the LibreTexts platform.

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