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.

Since you have to solve a separable equation, you must be particularly careful about the constant solutions.




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