First Order Differential Equation with Homogenous Coefficients

First Order Differential Equation with Homogenous Coefficients

Definition of a Homogenous Function

Note (jhobbs): Homogenous functions are not specific to differential equations. They have an algebraic definition that can apply to any functions, but it is the first place I’ve encountered them.

Definition 7.1 homogenous of order n Let $z = f(x,y)$ define $z$ as a function of $x$ and $y$ in a region $R$. The function $f(x,y)$ is said to be homogenous of order n if it can be written as

\[f(x,y) = x^ng(u), \tag{7.11}\]

where $u = y/x$ and $g(u)$ is a function of $u$; or alternatively if it can be written as

\[f(x,y) = y^nh(h) \tag{7.12}\]

where $u = x/y$ and $h(u)$ is a function of $u$.

Comment 7.15 Alternate definition An alternate definition of a homogenous function is the following. A function $f(x,y)$ is said to be homogenous of order n if

\[f(tx,ty) = t^n(x,y), \tag{7.16}\]

where $t > 0$ and $n$ is a constant.

Note (jhobbs): Homogenous equations are generally not separated/separable.

Solution of a Differential Equation in Which the Coeffecients of $dx$ and $dy$ Are Each Homogenous Functions of the Same Order

Definition 7.2 The differential equation

\[P(x,y)dx + Q(x,y)dy = 0, \tag{7.3}\]

where $P(x,y)$ and $Q(x,y)$ are each homogenous functions of order $n$ is called a first order differential equation with homogenous coefficients.

jmh: Do $P(x,y)$ and $Q(x,y)$ need to be of the order $n$ and have the same factor, either $x^n$ or $y^n$, or can they have different factors of the same order? In this book, I’ve only seen where they have the same factor.

The following substitution in (7.3) will always lead to a differential equation in $x$ and $u$ in which the variables are separable and hence solvable for $u$ by lesson 6.

\[y = ux, dy = udx + xdu \tag{7.31}\]

Theorem 7.32 If the coeffecients in (7.3) are each homogenous functions of order n, then the substition in it of (7.31) will lead to an equation in which the variables are separable.

Proof: see the book.

We could also use, instead of (7.31), the following substitution

\[x = uy, dx = udy + ydu \tag{7.33}\]

The one to use depends on the situation - sometimes one leads to an easier to compute result than the other, though the result will always be the same.