Mathematics · Class 12 · Differential Equations · Term 3

Solving Homogeneous Differential Equations

Identify homogeneous differential equations and solve them by making the substitution y = vx, which transforms them into variable separable form.

TL;DR:Let's tackle a special type of differential equation that looks tricky but has a secret key to unlock its solution.

• CBSE Learning Outcomes
• NCERT/CBSE Class 12 Mathematics: Chapter 9 - Differential Equations

About This Topic

This topic, Solving Homogeneous Differential Equations, is a crucial component of the Differential Equations chapter in the Class 12 mathematics curriculum, as prescribed by the NCERT and followed by CBSE and other state boards. It builds directly upon the foundational concept of solving differential equations by the variable separable method. The core idea is to introduce students to a specific type of first-order, first-degree differential equation that is not immediately separable but can be transformed into a separable one through a clever substitution. The form `dy/dx = F(y/x)` is the hallmark of such equations.

The pedagogical approach should focus on two main skills: identification and transformation. First, students must learn to test if a differential equation is homogeneous by checking if the functions involved are homogeneous functions of the same degree. This involves the `f(λx, λy) = λ^n f(x, y)` test. Second, they must master the substitution `y = vx`, which, through the application of the product rule for differentiation, `dy/dx = v + x(dv/dx)`, elegantly converts the equation into a form where the variables `v` and `x` can be separated. This topic serves as a bridge, reinforcing integration techniques and algebraic manipulation, while preparing students for the next method: solving linear differential equations.

Key Questions

1. Identify whether a given function or differential equation is homogeneous, explaining the criteria used.
2. Explain why the substitution y = vx is effective in transforming a homogeneous differential equation into a variable separable one.
3. Analyse the complete solution process for a first-order homogeneous differential equation, from identification to final integration.

Learning Objectives

• Define a homogeneous function and a homogeneous differential equation.
• Test a given first-order differential equation for homogeneity.
• Apply the substitution `y = vx` and `dy/dx = v + x(dv/dx)` to transform a homogeneous differential equation.
• Solve the transformed variable separable equation through integration.
• Express the final general solution in terms of the original variables `x` and `y`.

Key Vocabulary

Homogeneous Function	A function `F(x, y)` is called homogeneous of degree `n` if `F(λx, λy) = λ^n F(x, y)` for any non-zero constant λ.
Homogeneous Differential Equation	A first-order differential equation that can be expressed in the form `dy/dx = F(y/x)` or `dx/dy = G(x/y)`.
Substitution	The process of replacing a variable or an expression with another variable or expression to simplify an equation.
Variable Separable Form	A differential equation that can be written in the form `f(y)dy = g(x)dx`, where all `y` terms are on one side and all `x` terms are on the other.

Watch Out for These Misconceptions

Common Misconception

Students forget to substitute for `dy/dx` correctly. They substitute `y = vx` but forget that `dy/dx` becomes `v + x(dv/dx)` via the product rule.

What to Teach Instead

Emphasise that since `y` is being replaced by a product of two functions of `x` (since `v` is a function of `x`), the product rule for differentiation is essential. Always write the substitution pair together: `y = vx` and `dy/dx = v + x(dv/dx)`.

Common Misconception

The final answer is left in terms of `v` and `x` instead of the original variables `y` and `x`.

What to Teach Instead

Remind students that the original problem was about the relationship between `y` and `x`. The variable `v` is just a temporary tool. The final step must always be to substitute back `v = y/x` to get the solution in its proper form.

Common Misconception

Confusion between a homogeneous function and a homogeneous equation. A student might see a term like `sin(y/x)` and assume the whole equation is homogeneous without checking other terms.

What to Teach Instead

Clarify that a differential equation of the form `M(x,y)dx + N(x,y)dy = 0` is homogeneous only if both `M(x,y)` and `N(x,y)` are homogeneous functions of the *same degree*.

Active Learning Ideas

Collaborative Problem-Solving

Homogeneity Spotting

Provide a worksheet with a mix of ten differential equations. In pairs, students must identify which are homogeneous and justify their choice by showing that the equation can be written in the form `dy/dx = F(y/x)`.

15 min·Pairs

Collaborative Problem-Solving

Substitution Relay Race

In small groups, each student is responsible for one step of the solution process: one identifies homogeneity, another performs the `y=vx` substitution, a third separates the variables, and the last one integrates.

25 min·Small Groups

Collaborative Problem-Solving

Solve and Swap

Each group solves a unique homogeneous differential equation on a chart paper. Afterwards, groups rotate to a new station to check and understand the solution presented by another group.

30 min·Small Groups

Real-World Connections

• In economics, modelling relationships between production output, capital, and labour using Cobb-Douglas production functions can lead to homogeneous differential equations.
• In physics, determining the path of an object moving through a medium where resistance is a function of the velocity's components.
• In population studies, modelling the ratio of two competing species where their growth rates depend on their relative population sizes.
• In chemistry, analysing the rate of certain reactions where the concentrations of reactants are interdependent in a homogeneous way.
• In geometry, finding the equation of a curve whose tangent at any point has a slope that is a function of the ratio `y/x`.

Assessment Ideas

Exit Ticket

Exit Ticket: Give students a homogeneous differential equation and ask them to only perform the substitution and rearrange it into variable separable form, without solving it completely. This checks the key transformation step.

Peer Assessment

Peer Review: Students solve a problem in pairs and then exchange their notebooks with another pair to check for correctness, focusing on the substitution and the final back-substitution step.

Quick Check

A short quiz or a section in the unit test with 2-3 problems requiring full solutions, including one with a given initial condition to find a particular solution.

Frequently Asked Questions

Why do we use the substitution `y = vx` specifically for these equations?

This substitution works because in a homogeneous equation, all terms have the same degree. When you substitute `y = vx`, the powers of `x` in the numerator and denominator cancel out, leaving an expression that only depends on `v`. This allows the equation to be rearranged into a variable separable form.

Can we use `x = vy` instead?

Yes, absolutely. The substitution `x = vy` is useful if the differential equation is easier to express in the form `dx/dy = G(x/y)`. The procedure is analogous: you would differentiate `x` with respect to `y` using the product rule.

What happens if I try to use `y = vx` on a non-homogeneous equation?

If the equation is not homogeneous, the substitution will not lead to the cancellation of `x` terms. You will be left with a more complicated differential equation in `v` and `x` that you likely cannot solve by separating variables.