Question

If \( u = x^3 \) and \( v = y^2 \) transform the differential equation \( 3x^5 dx - y(y^2 - x^3) dy = 0 \) to \( \dfrac{dv}{du} = \dfrac{\alpha u}{2(u - v)} \), then \( \alpha \) is
 

• A. 4
• B. 2
• C. -2
• D. -4

Hint:

Always convert \( dx, dy \) correctly using \( du, dv \) and simplify before substituting back.

Answer 1

The Correct Option is D

Step 1: Write given equation.
\( 3x^5 dx - y(y^2 - x^3) dy = 0. \)

Step 2: Substitute \( u = x^3, v = y^2. \)
Then \( du = 3x^2 dx, \, dv = 2y dy. \)

Step 3: Express \( dx \) and \( dy \).
\( dx = \dfrac{du}{3x^2}, dy = \dfrac{dv}{2y}. \)

Step 4: Substitute in given equation.
\( 3x^5 \dfrac{du}{3x^2} - y(y^2 - x^3)\dfrac{dv}{2y} = 0. \)
Simplify: \( x^3 du - \dfrac{1}{2}(y^2 - x^3) dv = 0. \)

Step 5: Substitute \( u = x^3, v = y^2. \)
\( u\, du - \dfrac{1}{2}(v - u) dv = 0. \)

Step 6: Rearrange.
\( \dfrac{dv}{du} = \dfrac{2u}{u - v} = \dfrac{\alpha u}{2(u - v)} $\Rightarrow$ \alpha = 4. \)

Final Answer: \( \alpha = 4. \)