Question

The solution of the differential equation \( (kx - y^2) dy = (x^2 - ky) dx \) is:

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

Hint:

Always look for the combination $(y dx + x dy)$ in differential equations; it is the exact differential of the product $xy$, allowing for immediate integration.

Answer 1

The Correct Option is A

Concept: A differential equation of the form \( M(x, y) dx + N(x, y) dy = 0 \) is exact if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). Alternatively, some equations can be solved by rearranging terms into total differentials.

Step 1: Rearrange the equation.
Rearrange the equation to the form \( M dx + N dy = 0 \): \[ (x^2 - ky) dx - (kx - y^2) dy = 0 \] \[ (x^2 - ky) dx + (y^2 - kx) dy = 0 \]

Step 2: Expand and group similar terms.
\[ x^2 dx - ky dx + y^2 dy - kx dy = 0 \] \[ x^2 dx + y^2 dy - k(y dx + x dy) = 0 \]

Step 3: Identify total differentials and integrate.
Recall that \( d(xy) = x dy + y dx \). Substituting this: \[ x^2 dx + y^2 dy - k \cdot d(xy) = 0 \] Integrating each term: \[ \int x^2 dx + \int y^2 dy - k \int d(xy) = C_1 \] \[ \frac{x^3}{3} + \frac{y^3}{3} - kxy = C_1 \] Multiplying the entire equation by 3: \[ x^3 + y^3 - 3kxy = 3C_1 \] \[ x^3 + y^3 = 3kxy + C \]