Q.A homogeneous differential equation of the form $\dfrac{dx}{dy} = h\left(\dfrac{x}{y}\right)$ can be solved by making the substitution (A) $y = vx$ (B) $v = yx$ (C) $x = vy$ (D) $x = v$

Concept: Homogeneous Differential Equation — an equation of the form $\frac{dx}{dy} = h\left(\frac{x}{y}\right)$ is homogeneous in $x$ and $y$ because the right-hand side depends only on the ratio $\frac{x}{y}$.

Reasoning:

1. To solve, we substitute $x = v y$, where $v$ is a new function of $y$. This gives $\frac{dx}{dy} = v + y \frac{dv}{dy}$.
2. The equation becomes $v + y \frac{dv}{dy} = h(v)$, which is separable in $v$ and $y$. …

Why the substitution $x = vy$?

A differential equation is called homogeneous if it can be written in the form

The key idea: the right-hand side depends only on the ratio $\frac{x}{y}$, not on $x$ and $y$ separately. To exploit this, we introduce a new variable $v$ that is that ratio:

But then $x = vy$. This is the substitution we use.

Why not $y = vx$? If we set $y = vx$, then $\frac{x}{y} = \frac{1}{v}$, which is fine — but the equation is written with $x$ as the dependent variable and $y$ as the independent variable. The form $\frac{dx}{dy} = h(x/y)$ tells us to treat $y$ as the independent variable. So we want to express $x$ in terms of $y$ and a new variable. That’s exactly $x = vy$.

Step-by-step solution

1. Identify the form.

   The given equation is $\frac{dx}{dy} = h\left(\frac{x}{y}\right)$. This is homogeneous in $x$ and $y$, with $y$ as the independent variable.

2. Make the substitution.

   Let $x = vy$, where $v$ is a function of $y$. Then differentiate with respect to $y$:

3. Replace into the original equation. The right-hand side becomes $h\left(\frac{vy}{y}\right) = h(v)$. So we have:

4. Separate variables. …

Method: Choosing the correct substitution for a homogeneous equation

For a multiple-choice or reasoning question, the skill is matching the substitution to the form in which the equation is written, not memorising one rule.

Steps

Step 1: Read which variable is the independent one.

If the derivative is $\dfrac{dy}{dx}$, then $x$ is independent; if it is $\dfrac{dx}{dy}$, then $y$ is independent.

Step 2: Look at the ratio inside the function.

A homogeneous equation is $\dfrac{dy}{dx}=F\!\left(\dfrac{y}{x}\right)$ or $\dfrac{dx}{dy}=h\!\left(\dfrac{x}{y}\right)$. Introduce the new variable as that exact ratio.

Step 3: Solve for the dependent variable. …

Common Mistakes

Mistake 1: Reflexively choosing $y=vx$ for every homogeneous equation.

Why it's wrong: the equation here is $\dfrac{dx}{dy}=h\!\left(\dfrac{x}{y}\right)$ with $y$ independent, so the ratio to replace is $\dfrac{x}{y}$, giving $x=vy$. Blindly using $y=vx$ leaves $x/y=1/v$ and a messier equation. Correct approach: match the substitution to the ratio actually present. …