Ordinary Differential Equations/Homogeneous x and y

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Not to be confused with homogeneous equations, an equation homogeneous in x and y of degree n is an equation of the form

F(x,y,y')=0

Such that

.

Then the equation can take the form

Which is essentially another in the form

.

If we can solve this equation for y', then we can easily use the substitution method mentioned earlier to solve this equation. Suppose, however, that it is more easily solved for ,

So that

.

We can differentiate this to get

y'=f(y')+xf'(y')y

Then re-arranging things,

So that upon integrating,

We get

Thus, if we can eliminate y' between two simultaneous equations

and

,

then we can obtain the general solution..

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Homogeneous Ordinary Differential EquationsEdit

A function P is homogeneous of order   if  . A homogeneous ordinary differential equation is an equation of the form P(x,y)dx+Q(x,y)dy=0 where P and Q are homogeneous of the same order.

The first usage of the following method for solving homogeneous ordinary differential equations was by Leibniz in 1691. Using the substitution y=vx or x=vy, we can make turn the equation into a separable equation.

 
 
 

Now we need to find v':

 

Plug back into the original equation

 
 
Solve for v(x), then plug into the equation of v to get y
 

Again, don't memorize the equation. Remember the general method, and apply it.

Example 2Edit

 

Let's use  . Solve for y'(x,v,v')

 
 

Now plug into the original equation

 
 
 

Solve for v

 
 
 
 
 
 
 

Plug into the definition of v to get y.

 
 
 

We leave it in   form, since solving for y would lose information.

Note that there should be a constant of integration in the general solution. Adding it is left as an exercise.

Example 3Edit

 

Lets use   again. Solve for  

 
 

Now plug into the original equation

 
 
 

Solve for v:

 
 
 

Use the definition of v to solve for y.

 
 

An equation that is a function of a quotient of linear expressionsEdit

Given the equation  ,

We can make the substitution x=x'+h and y=y'+k where h and k satisfy the system of linear equations:

 
 

Which turns it into a homogeneous equation of degree 0: