In the prior section we looked at Bernoulli Equations and noticed that in order to solve them we required to use the substitution v = y1-n. By using this substitution we were capable to convert the differential equation in a form which we could deal along with but, linear in this case. In this section we need to see a couple of other substitutions which can be used to reduce several differential equations down to a solvable form.
The first substitution we'll take a seem at will need the differential equation to be in the create,
y' = F(y/x)
First order differential equations which can be written in this form are termed as homogeneous differential equations. Remember that we will generally have to do several rewriting in order to place the differential equation in the exact form.
Once we have verified as the differential equation is a homogeneous differential equation and we've gotten this written in the exact form we will use the subsequent substitution.
n (x) = y/x
We can then rewrite this as,
y = xn
And after that remembering that both y and v are functions of x we can utilize the product rule to calculate,
y′ = n + xn′
In this substitution the differential equation is like,
n + xn′ = F(n)
⇒ xn′ = F(n) - n
⇒ dv/ F(v) - v = dx/x
When we can notice with a small rewrite of the new differential equation we will have a separable differential equation after the substitution.