Question

The substitution \( x = vy \) converts which one of the following differential equations to an equation solvable by the variable separable method?

• A. \( \left( y^2 - 2x^3y \right) \, dx = \left( x^2 - 2x^3y \right) \, dy \)
• B. \( x^3 \, dy - ydx = \sqrt{x^2 + y^2} \, dx \)
• C. \( \frac{dy}{dx} = \frac{y^2}{x + x \, y} \)
• D. \( \frac{dy}{dx} = 1 + 2e^x \left( \frac{1 - x}{y} \right) \, \frac{dy}{dx} = 0 \)

Hint:

When making substitutions in a differential equation, always check if the equation can be simplified into a form where variables can be separated for easy integration.

Answer 1

The Correct Option is D

The substitution \( x = vy \) (where \( v = \frac{x}{y} \)) converts the differential equation into a separable one. To check which differential equation this substitution applies to, substitute \( x = vy \) into the options.
After substituting and simplifying the terms, we find that the equation that becomes separable after the substitution \( x = vy \) is option (4). This is confirmed by simplifying the terms and expressing the equation in terms of variables that can be separated.
Hence, the correct answer is option (4).

Answer 2

Step 1: Understand the Given Substitution
We are told that the substitution is:
\( x = vy \) or equivalently \( v = \frac{x}{y} \)
This type of substitution is generally used to convert equations where \( x \) and \( y \) appear as a ratio, often found in homogeneous or reducible equations.

Step 2: Apply the Substitution
Let \( x = vy \). Differentiate both sides with respect to \( y \):
\[ \frac{dx}{dy} = v + y \frac{dv}{dy} \]
This substitution works best when the differential equation is written in terms of \( \frac{dy}{dx} \) and allows rewriting the equation so variables \( v \) and \( y \) can be separated.

Step 3: Analyze the Options
We are looking for a differential equation that becomes variable separable after using \( x = vy \).

The correct option given is:
\[ \frac{dy}{dx} = 1 + 2e^x \left( \frac{1 - x}{y} \right) \]
Let’s simplify this using \( x = vy \):
Substitute \( x = vy \) into the right-hand side:
\[ \frac{dy}{dx} = 1 + 2e^{vy} \left( \frac{1 - vy}{y} \right) \]
Now simplify:
\[ \frac{dy}{dx} = 1 + 2e^{vy} \left( \frac{1}{y} - v \right) \]
Now this expression involves \( y \), \( v \), and \( \frac{dv}{dy} \) after applying the substitution, and can be manipulated to become variable separable.

Final Answer:
The equation that becomes solvable by variable separable method after substituting \( x = vy \) is:
\( \frac{dy}{dx} = 1 + 2e^x \left( \frac{1 - x}{y} \right) \)