Homogeneous ODEs y′ = F(y/x)
About this app
ODEs y′ = F(y/x) in the direction field: isoclines are lines through the origin, a dilation maps solutions to solutions (C′ = C − ln λ), and zooming towards the origin leaves the field unchanged. Five-step worked solution with numbers: recognise, u = y/x, separate x·u′ = F(u) − u, substitute back, initial value. Five examples up to the logarithmic spiral, invariant lines F(u) = u, and a “homogeneous?” test with a ray check.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
What a homogeneous ODE is
A differential equation y′ = f(x, y) is called homogeneous (of similarity type) if the right-hand side depends only on the ratio y/x, i.e. y′ = F(y/x). Equivalently, f(tx, ty) = f(x, y) for all t ≠ 0. It is solved with the substitution u = y/x: from y = x·u the product rule gives y′ = u + x·u′, and the equation becomes x·u′ = F(u) − u. That is separable. After integrating, substitute back y = x·u and fix the constant from the initial value.
What the app shows
The app draws the direction field of five examples, among them y′ = (x + y)/x with solution y = x·(ln|x| + C) and y′ = (x + y)/(x − y) with logarithmic spirals. A worked solution leads in five steps from recognising to the initial value, with numbers inserted. The scene shows the matching isoclines: on every line through the origin the slope is the same. A dilation about the origin maps the solution through P onto the solution through λ·P, and when zooming towards the origin the field stays exactly the same.
What you can try
Drag the initial point P or tap into the field, change the dilation factor λ and watch the dashed, dilated curve coincide with a solution. Start the zoom ride: the field stays put while the curves turn into one another. Go through steps 1 to 5 one by one, and for y′ = y²/x² look at the invariant lines y = 0 and y = x that get lost when dividing. In the test you decide for 14 equations whether they are homogeneous and see the ray check in the field.
Frequently asked questions
- How do you recognise a homogeneous ODE?
- Replace x by tx and y by ty. If t cancels completely, i.e. f(tx, ty) = f(x, y), the right-hand side depends only on y/x. Example: (x + y)/x becomes (tx + ty)/(tx) = (x + y)/x. For x + y, however, t·(x + y) remains.
- How do you solve a homogeneous ODE y′ = F(y/x)?
- Set u = y/x, so y = x·u and y′ = u + x·u′. Then x·u′ = F(u) − u. Separate this equation (du/(F(u) − u) = dx/x), integrate both sides, substitute back y = x·u and determine the constant from the initial value. The substitution requires x ≠ 0.
- Why are the isoclines of a homogeneous ODE lines through the origin?
- On a line y = kx through the origin, y/x = k is constant, and so is the slope F(k). That is why a dilation about the origin maps solution curves to solution curves. Lines through the origin with F(k) = k are solutions themselves; they get lost when dividing by F(u) − u.
- What can I adjust in the app?
- You pick one of five examples, drag the initial point, set the dilation factor and the zoom, go through the worked solution step by step and show isoclines and the family of solutions. A test asks for 14 equations whether they are homogeneous. An explain mode walks through everything.
Subject: Differential equations
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- Separation of variables
- Worlds of differential equations
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