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⬅ Back to Differential Equations
📘 Key Concepts: Homogeneous Equations

A first-order DE is homogeneous if it can be written as $\dfrac{dy}{dx} = F\!\left(\dfrac{y}{x}\right)$. Use the substitution $v = \dfrac{y}{x}$, so $y = vx$ and:

$$\frac{dy}{dx} = v + x\frac{dv}{dx}$$

This converts the equation into a separable DE in $v$ and $x$.

Steps:

  1. Verify the equation is homogeneous (all terms same degree, or rewrite as $dy/dx = F(y/x)$).
  2. Substitute $v = y/x$, i.e. $y = vx$, $dy/dx = v + x\,dv/dx$.
  3. Separate and integrate.
  4. Back-substitute $v = y/x$ and apply initial conditions.

Problem 1: Basic Homogeneous Equation

Solve $(x + y)\,dx - x\,dy = 0$, $y(1) = 0$.

DE – Homogeneous – Problem 1 – Setup DE – Homogeneous – Problem 1 – Diagram DE – Homogeneous – Problem 1 – Diagram
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Problem 2: Homogeneous with Initial Condition

Solve $(y^2 - xy)\,dx + x^2\,dy = 0$, $y(1) = 1$.

DE – Homogeneous – Problem 2 – Setup DE – Homogeneous – Problem 2 – Diagram DE – Homogeneous – Problem 2 – Diagram
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Problem 3: Homogeneous with Irrational Term

Solve $x\,dy - y\,dx = \sqrt{x^2 + y^2}\,dx$, $y(1) = 0$.

DE – Homogeneous – Problem 3 – Setup DE – Homogeneous – Problem 3 – Diagram DE – Homogeneous – Problem 3 – Diagram
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Problem 4: Homogeneous — General Solution

Solve $\dfrac{dy}{dx} = \dfrac{x + y}{x - y}$, $y(1) = 0$. Express the answer in implicit form.

DE – Homogeneous – Problem 4 – Setup DE – Homogeneous – Problem 4 – Diagram DE – Homogeneous – Problem 4 – Diagram
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