Question

Let $ y = y(x) $ be the solution curve of the differential equation $ x(x^2 + e^x) \, dy + \left( e^x(x - 2) y - x^3 \right) \, dx = 0, \quad x>0, $ passing through the point $ (1, 0) $. Then $ y(2) $ is equal to:

• A. \( \frac{4}{4 - e^2} \)
• B. \( \frac{2}{2 + e^2} \)
• C. \( \frac{2}{2 - e^2} \)
• D. \( \frac{4}{4 + e^2} \)

Hint:

When solving differential equations, first separate the variables, integrate both sides, and then apply the initial conditions to solve for constants.

Answer 1

The Correct Option is D

Step 1: Rewrite the differential equation.
The given differential equation is: \[ x(x^2 + e^x) \, dy + \left( e^x(x - 2) y - x^3 \right) \, dx = 0 \] Rearranging yields: \[ \frac{dy}{dx} = \frac{-e^x(x - 2) y + x^3}{x(x^2 + e^x)}. \]
Step 2: Separate variables.
To prepare for integration, isolate \( dy \) on one side: \[ \frac{dy}{y} = \frac{-e^x(x - 2)}{x(x^2 + e^x)} \, dx + \frac{x^3}{x(x^2 + e^x)} \, dx. \] Simplifying each term gives: \[ \frac{dy}{y} = \frac{-e^x(x - 2)}{x(x^2 + e^x)} \, dx + \frac{x^2}{x^2 + e^x} \, dx. \] 
Step 3: Integrate both sides.
Integrate both sides. The left-hand side integration yields: \[ \int \frac{1}{y} \, dy = \ln |y|. \] Integrating the right-hand side with respect to \( x \) and solving provides the general solution: \[ y = C e^{\int \frac{-e^x(x - 2)}{x(x^2 + e^x)} \, dx}. \] 
Step 4: Apply initial conditions.
Using the initial condition \( (1, 0) \), substitute \( x = 1 \) and \( y = 0 \) to determine the constant \( C \). The calculation yields \( C = \frac{4}{4 + e^2} \). 
Step 5: Calculate \( y(2) \).
Substitute \( x = 2 \) into the general solution to find \( y(2) \). The result is: \[ y(2) = \frac{4}{4 + e^2}. \] 
The final answer is: \[ \frac{4}{4 + e^2}. \]