Ordinary Differential Equations/Homogeneous x and y

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.. Homogeneous Ordinary Differential Equations 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 2 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. Example 3 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 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: