Question

If \[ x^a \frac{dy}{dx} = y^\beta \left( \gamma \log x + \delta y + 1 \right) \] is a homogeneous differential equation, then

• A. \( \alpha = \beta \) and \( \gamma = \delta \)
• B. \( \alpha = \beta \) and \( \gamma \neq \delta \)
• C. \( \alpha \neq \beta \) and \( \gamma = \delta \)
• D. \( \alpha \neq \beta \) and \( \gamma \neq \delta \)

Hint:

For homogeneous differential equations, ensure that the powers of variables in each term match, as this is a key condition for homogeneity.

Answer 1

The Correct Option is A

We are given a homogeneous differential equation of the form: \[ x^a \frac{dy}{dx} = y^\beta \left( \gamma \log x + \delta y + 1 \right) \] Step 1: For the equation to be homogeneous, the powers of \( x \) and \( y \) in both terms must balance out, implying that \( \alpha = \beta \) and \( \gamma = \delta \). Thus, the correct relationship between the constants is \( \alpha = \beta \) and \( \gamma = \delta \). % Final Answer The values of \( \alpha \), \( \beta \), \( \gamma \), and \( \delta \) must satisfy \( \alpha = \beta \) and \( \gamma = \delta \).

Answer 2

Step 1: Understand the given differential equation
The equation is:
\[ x^\alpha \frac{dy}{dx} = y^\beta \left( \gamma \log x + \delta y + 1 \right) \]

Step 2: Condition for homogeneity
A differential equation is homogeneous if it can be expressed in the form where the right side is a function of \(\frac{y}{x}\) only, or equivalently, if all terms are of the same degree when \(x\) and \(y\) are scaled.

Step 3: Analyze each term for homogeneity
- The term \(x^\alpha \frac{dy}{dx}\) involves \(x\) raised to power \(\alpha\).
- The term \(y^\beta\) is \(y\) raised to power \(\beta\).
- The term \(\gamma \log x\) involves a logarithm, which is not homogeneous by itself.
- The term \(\delta y\) is linear in \(y\).
- The constant term \(1\) is of degree zero.

Step 4: Ensuring homogeneity
To make the equation homogeneous:
- The powers of \(x\) and \(y\) on both sides must match, so \(\alpha = \beta\).
- The logarithmic term \(\gamma \log x\) and the linear term \(\delta y\) must scale the same way under the transformation \(x \to kx, y \to ky\).
Since \(\log x\) does not scale as a power function, for the entire expression to be homogeneous, the coefficients of \(\log x\) and \(y\) must be related such that the combination behaves like a homogeneous function.

Step 5: Conclusion
The only way to satisfy homogeneity here is if:
\[ \alpha = \beta \quad \text{and} \quad \gamma = \delta \].
This makes the terms consistent in degree and allows the equation to be homogeneous.