Question

The general solution of the differential equation \(\frac{dy}{dx} = \frac{x^2}{y^2}\) is :

• A. \(x^2 + y^2 = c\)
• B. \(x^2 - y^2 = c\)
• C. \(x^3 + y^3 = c\)
• D. \(x^3 - y^3 = c\)

Hint:

To verify the solution, differentiate \(x^3 - y^3 = c\) implicitly with respect to \(x\):
\[ 3x^2 - 3y^2 \frac{dy}{dx} = 0 \implies 3y^2 \frac{dy}{dx} = 3x^2 \implies \frac{dy}{dx} = \frac{x^2}{y^2} \]
This confirms our solution is correct.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a first-order, first-degree ordinary differential equation.
Since the terms involving \(x\) and \(y\) can be completely separated on opposite sides of the equation, we can solve it using the method of Separation of Variables.
Key Formula or Approach:
Write the differential equation in the form:
\[ g(y) \, dy = f(x) \, dx \]
Then, integrate both sides:
\[ \int g(y) \, dy = \int f(x) \, dx + c' \]

Step 2: Detailed Explanation:
Let us start with the given differential equation:
\[ \frac{dy}{dx} = \frac{x^2}{y^2} \]
Multiply both sides by \(y^2 \cdot dx\) to separate the variables:
\[ y^2 \, dy = x^2 \, dx \]
Now, integrate both sides:
\[ \int y^2 \, dy = \int x^2 \, dx \]
Using the standard power rule of integration \(\int u^n \, du = \frac{u^{n+1}}{n+1}\):
\[ \frac{y^3}{3} = \frac{x^3}{3} + c' \]
Where \(c'\) is the constant of integration.
Multiply the entire equation by 3 to eliminate the denominators:
\[ y^3 = x^3 + 3c' \]
Rearranging the terms:
\[ x^3 - y^3 = -3c' \]
Let us define a new arbitrary constant \(c = -3c'\).
The equation becomes:
\[ x^3 - y^3 = c \]
This is the general solution of the given differential equation.
This matches Option (D).

Step 3: Final Answer:
The correct option is (D).