Question

Let \( y = y(x) \) be the solution of the differential equation \[ \left( xy - 5x^2 \sqrt{1 + x^2} \right) dx + (1 + x^2) dy = 0, \quad y(0) = 0. \] Then \( y(\sqrt{3}) \) is equal to:

• A. \( \frac{5\sqrt{3}}{2} \)
• B. \( \sqrt{\frac{14}{3}} \)
• C. \( 2\sqrt{2} \)
• D. \( \sqrt{\frac{15}{2}} \)

Hint:

To solve first-order differential equations, isolate \( dy \) on one side, and then integrate with respect to \( x \). Don't forget to apply the initial condition to find the constant of integration.

Answer 1

The Correct Option is A

Step 1: The given differential equation is: \[ \left( xy - 5x^2 \sqrt{1 + x^2} \right) dx + (1 + x^2) dy = 0. \] We can rearrange the equation as: \[ (1 + x^2) dy = -\left( xy - 5x^2 \sqrt{1 + x^2} \right) dx. \] Next, we divide by \( (1 + x^2) \): \[ dy = -\frac{xy - 5x^2 \sqrt{1 + x^2}}{1 + x^2} dx. \] This is a first-order differential equation. We solve this equation by integrating both sides. After integrating and applying the initial condition \( y(0) = 0 \), we obtain the solution for \( y(x) \). 
Step 2: Now, substitute \( x = \sqrt{3} \) into the solution to find \( y(\sqrt{3}) \): \[ y(\sqrt{3}) = \frac{5\sqrt{3}}{2}. \] Thus, the value of \( y(\sqrt{3}) \) is \( \frac{5\sqrt{3}}{2} \).