The differential equation
is homogeneous if the function f(x,y) is homogeneous, that is-
Check that the functions
In order to solve this type of equation we make use
of a substitution (as we did in case of Bernoulli equations). Indeed, consider
the substitution . If f(x,y) is
homogeneous, then we have
Since y' = xz' + z, the equation (H) becomes
which is a
separable equation. Once solved, go back to the old
variable y via the equation y = x z.
Let us summarize the steps to follow:
Since you have to solve a separable equation, you must be
particularly careful about the constant solutions.
- Recognize that your equation is an homogeneous equation; that is, you
need to check that f(tx,ty)= f(x,y), meaning that
f(tx,ty) is independent of the variable t;
- Write out the substitution z=y/x;
- Through easy differentiation, find the new equation satisfied
by the new function z.
You may want to remember the form of the new equation:
- Solve the new equation (which is always
separable) to find z;
- Go back to the old function y through the substitution y =
- If you have an IVP, use the initial condition to find the
Example: Find all the solutions of
Solution: Follow these steps:
- It is easy to check that is homogeneous;
- Consider ;
- We have
which can be rewritten as
This is a separable equation.
If you don't get a separable equation
at this point, then your equation is not
homogeneous, or something went wrong along the way.
- All solutions are given implicitly by
- Back to the function y, we get
Note that the implicit equation can be rewritten as
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