Homogeneous Differential Equations

Precise definition

A first-order equation is homogeneous in the substitution sense when it can be written $y'=F(y/x)$, or when $M(x,y)dx+N(x,y)dy=0$ has $M,N$ homogeneous of the same degree. Setting $v=y/x$ so $y=vx$ and $y'=v+xv'$ converts it to a separable equation.

Notation and mathematical language

This meaning differs from a homogeneous linear equation, whose forcing is zero. Here $v(x)=y(x)/x$, so the method works on intervals with $x\ne0$. After substitution, $xv'=F(v)-v$ and, where the denominator is nonzero, $dv/[F(v)-v]=dx/x$.

Conceptual picture

The equation is unchanged along rays from the origin because $y/x$ is constant on each ray. The substitution tracks the evolving ray slope $v$. Roots of $F(v)-v=0$ give straight-line solutions $y=vx$ and can be lost during separation.

Conditions and key results

Scaling tests such as $M(tx,ty)=t^kM(x,y)$ identify homogeneous functions. The domain must exclude $x=0$ unless the original equation allows a separate extension. Dividing by $F(v)-v$ requires restoring constant-$v$ solutions.

A reliable strategy

1. Confirm dependence on $y/x$ or equal homogeneity degree and choose an interval with $x\ne0$.
2. Set $y=vx$ and replace $y'$ by $v+xv'$.
3. Find constant-$v$ solutions, then separate and integrate the remaining equation.
4. Return to $v=y/x$, apply initial data, and verify both the differential equation and interval.

Fully worked example

Interpretation and application

These equations model scale-invariant slopes and geometric families where only a ratio matters. Scale invariance is a property of the mathematical formulation; measurement offsets or fixed external scales can break it and require another model.

Common mistakes

Verification and reasonableness

• Substitute $y=xv$ into every term before simplifying.
• Differentiate the recovered $y(x)$ and compare directly with $F(y/x)$.
• Check whether the initial interval crosses $x=0$ or another logarithmic/domain boundary.

Practice

1. What is $y'$ when $y=xv(x)$?
2. What equation does $y'=F(y/x)$ become?
3. What special solutions require separate checking?

Further deduction

For a differential form, Euler's theorem helps verify homogeneity: if $M$ is differentiable and homogeneous of degree $k$, then $xM_x+yM_y=kM$. This test is optional but useful when scaling is not obvious. Equal degrees ensure the common power of $x$ cancels after $y=vx$, leaving a separable relationship between $v$ and $x$.

A scaling check can be performed directly on a slope function: $G(x,y)$ has degree zero when $G(tx,ty)=G(x,y)$ for nonzero $t$, so it depends only on the ratio $y/x$ on suitable regions. For example, $(x+y)/(x-y)=(1+y/x)/(1-y/x)$. Rewriting explicitly in the ratio confirms the substitution and reveals excluded rays such as $x=y$.

Related topics

• Separable Differential Equations
• Bernoulli Differential Equations
• Exact Differential Equations