Not to be confused with homogeneous equations, an equation homogeneous in x and y of degree n is an equation of the formSuch that
.
Then the equation can take the form
Which is essentially another in the form
.
If we can solve this equation for , 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
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..
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.
Let's use . Solve for
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.
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 expressions
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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: