Math 555: Differential Equations I |
A first order differential equation $\tfrac{dy}{dx}=f(x,y)$ is said to be homogeneous
if the right-hand side may be regarded as a function of $v = \tfrac{y}{x}$, so that $f(x,y)
= f(v)$. Such FODE can be made separable by the following procedure. Even if the DE is
already separable, this procedure usually simplifies the situation.
Introduce the change-of-variables $v = \tfrac{y}{x}$, so that $y = xv(x)$. Then by the
product rule,
$$
\frac{dy}{dx} = v + x\dfrac{dv}{dx}. \tag{1}
$$
Substituting into the original DE and solving for $\tfrac{dv}{dx}$ yields
$$
\frac{dv}{dx} = \frac{f(v) - v}{x}, \tag{2}
$$
a separable FODE for $v$.
The solutions to a homogeneous differential equation have interesting geometric properties.
Specifically, the solution curves are symmetric with respect to the origin (or they have
"rotational symmetry" about the origin).
We look at two typical examples below.
Example 1. $\displaystyle \ \ \frac{dy}{dx} = \frac{y-4x}{x-y}$
Let's begin by looking at the slope field.
Making the substitution $v = \tfrac{y}{x}$, this DE becomes
$$
v + x\frac{dv}{dx} = \frac{v-4}{1-v}.
$$
Solving for $\tfrac{dv}{dx}$ yields
$$
\frac{dv}{dx} = \frac{1}{x} \frac{v^2 - 4}{1 - v},
$$
which separates to give
$$
\frac{1-v}{v^2 - 4}\, dv = \frac{1}{x}\, dx.
$$
Now, integrating yields
$$
-\frac{3}{4}\ln(v+2) - \frac{1}{4}\ln(v-2) = \ln x + C.
$$
Finally, applying log rules, exponentiating, solving for $C$, and subbing back in
$\frac{y}{x}$ for $v$ yields
$$
y^4 + 4y^3x - 16yx^3 - 16x^4 = C.
$$
We conclude by plotting a few level curves on the slope field. Notice that the
solution curves are symmetric with respect to the origin.
Example 2. $\displaystyle \ \ \frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}$.
The substitution $v = \tfrac{y}{x}$ transforms the equation to
$$
\frac{dv}{dx} = \frac{1 + v^2}{x},
$$
which yields the solution
$$
y = x\tan\left(\ln x +C \right).
$$
The details are left as an exercise.
Again we plot the slope field and a few solution curves, and notice that they are symmetric with respect to the origin.
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