Homogeneous First-order Equations

There are two ways to view "homogeneous" first-order ordinary differential equations.

1. \(\frac{dy}{dx}=f(x,y)\) is a general first order ordinary differential equation.  If the function on the right, \(f(x,y)\), can be rewritten as a function of \(\frac{y}{x}\), e.g. \(f(x,y)=\frac{y-2x}{x}\) can be rewritten as \(g\left(\frac{y}{x}\right)=\frac{y}{x}-2\).  If we let \(v=\frac{y}{x}\), then we would have \(g(v)=v-2\).

In general, the substitution, \(v=\frac{y}{x}\) or \(y=vx\), will lead to a separable differential equation.

Since \(y=vx\) ,  differentiating both sides gives  \(\frac{dy}{dx}=\frac{dv}{dx}x+v\).  This new version of \(\frac{dy}{dx}\) becomes our new left side and \(g(v)\) becomes the right.




 The above version of our differential equation has been separated.

2. First-order linear equations, \(\displaystyle \frac{dy}{dx}+P(x)y=Q(x)\), are also separable if \(Q(x)\equiv 0\).





Leave a Response

You must be logged in to post a comment.