Question

Consider the following differential equations:
I: $y' = \frac{y+x}{x}$
II: $y' = \frac{x^2+y}{x^3}$
III: $y' = \frac{2xy}{y^2-x^2}$
Which of the statements below is true regarding these equations?
S1: Differential equations given by I and II are homogeneous differential equations.
S2: Differential equations given by II and III are homogeneous differential equations.
S3: Differential equations given by I and III are homogeneous differential equations.

• A. only S1 is valid
• B. both S1 and S2 are valid
• C. only S3 is valid
• D. only S2 is valid

Hint:

To spot homogeneous equations instantly without doing math, look at the exponents of each term!
In Eq. I: $y^1, x^1, x^1 \rightarrow$ matches. In Eq. III: $x^1y^1$ (total 2), $y^2, x^2 \rightarrow$ matches.
In Eq. II: $x^2$ and $y^1$ are mixed together in the numerator $\rightarrow$ instant mismatch!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question requires us to evaluate three different first-order ordinary differential equations and classify them to determine which are homogeneous differential equations.

Step 2: Key Formula or Approach:
A differential equation of the form $\frac{dy}{dx} = f(x,y)$ is classified as a homogeneous differential equation if $f(x,y)$ is a homogeneous function of degree zero.
This means that substituting $x \rightarrow \lambda x$ and $y \rightarrow \lambda y$ must satisfy:
$$f(\lambda x, \lambda y) = \lambda^0 f(x,y) = f(x,y)$$ Practically, this means every individual term within the numerator and the denominator expressions must have the exact same total degree (sum of powers of $x$ and $y$).

Step 3: Detailed Explanation:
Let's test each differential equation individually:

Equation I: $y' = \frac{y+x}{x}$
Degree of terms in numerator: $y$ has degree 1, $x$ has degree 1.
Degree of terms in denominator: $x$ has degree 1.
Since all terms in the fraction share an identical degree of 1, Equation I is a

homogeneous differential equation.

Equation II: $y' = \frac{x^2+y}{x^3}$
Degree of terms in numerator: $x^2$ has degree 2, while $y$ has degree 1.
Because the terms in the numerator have different degrees ($2 \neq 1$), the function is not homogeneous. Thus, Equation II is

not a homogeneous differential equation.

Equation III: $y' = \frac{2xy}{y^2-x^2}$
Degree of terms in numerator: $2xy$ has a total combined degree of $1 + 1 = 2$.
Degree of terms in denominator: $y^2$ has degree 2, and $x^2$ has degree 2.
Since every single term features a uniform total degree of 2, Equation III is a

homogeneous differential equation.
Evaluating our results: Equations I and III are homogeneous, which means statement

S3 is correct, while S1 and S2 are invalid.

Step 4: Final Answer:
Only statement S3 is valid, which corresponds perfectly to option (C).