Question

The solution(s) of the differential equation \[ \frac{dy}{dx} = (\sin 2x) y^{1/3} \text{satisfying} y(0) = 0 \text{is (are)} \]

• A. \( y(x) = 0 \)
• B. \( y(x) = -\frac{\sqrt{8}}{27} \sin^3 x \)
• C. \( y(x) = \frac{\sqrt{8}}{27} \sin^3 x \)
• D. \( y(x) = \frac{\sqrt{8}}{27} \cos^3 x \)

Hint:

When solving differential equations with power terms like \( y^{1/3} \), use separation of variables and apply initial conditions to find the specific solution.

Answer 1

The Correct Option is A, B, C

Step 1: Solving the differential equation.
We solve the differential equation \( \frac{dy}{dx} = (\sin 2x) y^{1/3} \) using separation of variables. First, rewrite the equation: \[ \frac{dy}{y^{1/3}} = (\sin 2x) dx \] Integrating both sides, we get: \[ 3y^{2/3} = \int (\sin 2x) dx \] The integral of \( \sin 2x \) is \( -\frac{1}{2} \cos 2x \), so: \[ 3y^{2/3} = -\frac{1}{2} \cos 2x + C \]

Step 2: Applying initial conditions.
Using \( y(0) = 0 \), we find that \( C = \frac{1}{2} \). Thus, the solution is: \[ y(x) = \left( \frac{\sqrt{8}}{27} \sin^3 x \right) \]

Step 3: Conclusion.
The correct answer is (C) \( y(x) = \frac{\sqrt{8}{27} \sin^3 x \)}.