A Differential Equation is an equation with a function and one or more of its derivatives:
differential equation dy/dx = 5xy
Example: an equation with the function y and its derivative dydx
Here we look at a special method for solving "Homogeneous Differential Equations"
Homogeneous Differential Equations
A first order Differential Equation is Homogeneous when it can be in this form:
dydx = F( yx )
We can solve it using Separation of Variables but first we create a new variable v = yx
v = yx which is also y = vx
And dydx = d (vx)dx = v dxdx + x dvdx (by the Product Rule)
Which can be simplified to dydx = v + x dvdx
Using y = vx and dydx = v + x dvdx we can solve the Differential Equation.
An example will show how it is all done:
Example: Solve dydx = x2 + y2xy
Can we get it in F( yx ) style?
Start with: x2 + y2xy
Separate terms: x2xy + y2xy
Simplify: xy + yx
Reciprocal of first term: ( yx )−1 + yx
Yes, we have a function of (y/x).
So let's go:
Start with: dydx = ( yx )−1 + yx
y = vx and dydx = v + x dvdx : v + x dvdx = v−1 + v
Subtract v from both sides: x dvdx = v−1
Now use Separation of Variables:
Separate the variables: v dv = 1x dx
Put the integral sign in front: ∫v dv = ∫ 1x dx
Integrate: v22 = ln(x) + C
Then we make C = ln(k): v22 = ln(x) + ln(k)
Combine ln: v22 = ln(kx)
Simplify: v = ±√(2 ln(kx))
Now substitute back v = yx
Substitute v = yx: yx = ±√(2 ln(kx))
Simplify: y = ±x √(2 ln(kx))
And we have the solution.
