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: Rearranging the equation

The given equation is: \[ \left( xy - 5x^2 \sqrt{1 + x^2} \right) dx + (1 + x^2) dy = 0. \] We can rearrange it as: \[ (1 + x^2) dy = - \left( xy - 5x^2 \sqrt{1 + x^2} \right) dx. \] \[ \Rightarrow \frac{dy}{dx} = - \frac{xy - 5x^2 \sqrt{1 + x^2}}{1 + x^2}. \]

Step 2: Simplifying the right-hand side

Simplify the expression on the right-hand side: \[ \frac{dy}{dx} = - \left( \frac{xy}{1 + x^2} - 5x \sqrt{1 + x^2} \right). \]

Step 3: Solving the equation

We can use the method of integration to solve this. Start by integrating the terms individually. The first term involves: \[ \int \frac{xy}{1 + x^2} \, dx. \] We can solve this using substitution or direct integration. After solving the equation, we obtain the solution for \( y(x) \).

Step 4: Apply initial condition

Using the given initial condition \( y(0) = 0 \), we determine the constant of integration.

Step 5: Find \( y(\sqrt{3}) \)

Finally, we substitute \( x = \sqrt{3} \) into the solution and obtain: \[ y(\sqrt{3}) = \frac{5\sqrt{3}}{2}. \]

Answer:

The value of \( y(\sqrt{3}) \) is \( \frac{5\sqrt{3}}{2} \).