The differential equation

is *homogeneous* if the function *f*(*x*,*y*) is homogeneous, that is-

Check that the functions

.

are homogeneous.

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:

**(1)**- 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*; **(2)**- Write out the substitution
*z*=*y*/*x*; **(3)**- Through easy differentiation, find the new equation satisfied
by the new function
*z*.

You may want to remember the form of the new equation: **(4)**- Solve the new equation (which is always
separable) to find
*z*; **(5)**- Go back to the old function
*y*through the substitution*y*=*x z*; **(6)**- If you have an IVP, use the initial condition to find the
particular solution.

**Example:** Find all the solutions of

**Solution:** Follow these steps:

**(1)**- It is easy to check that is homogeneous;
**(2)**- Consider ;
**(3)**- 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.

**(4)**- All solutions are given implicitly by
**(5)**- Back to the function
*y*, we getNote that the implicit equation can be rewritten as

**
**

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