Homogeneous Equations

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 = xz.

Example

Solve y'' + 3y - 4 = 0

The strategy is to search for a solution of the form y = e^rt. The reason for this is that long ago some geniuses figured this stuff out and it works.

Now calculate derivatives:

y' = re^rt
y'' = r²e^rt

Substituting into the differential equation gives

r²e^rt + 3(re^rt) - 4(e^rt)
= (r² + 3r - 4)e^rt = 0

Now divide by e^rt to get

r² + 3r - 4 = 0
(r - 1)(r + 4) = 0
r = 1, r = -4

We can conclude that two solutions are y₁ = e^t and y₂ = e^-4t

Now let L(y) = y'' + 3y' - 4. It is easy to verify that if y₁ and y₂ are solutions to L(y) = 0, then c₁y₁ + c₂y₂ is also a solution. More specifically we can conclude that y = c₁e^t + c₂e^-4t

Represents a two dimensional family (vector space) of solutions.