Question

Let \( y = y(x) \) be the solution of the differential equation \(\frac{dy}{dx} = 2x(x + y)^3 - x(x + y) - 1, \quad y(0) = 1.\)Then,  \(\left( \frac{1}{\sqrt{2}} + y\left(\frac{1}{\sqrt{2}}\right) \right)^2\)equals:

• A. \( \frac{4}{4 + \sqrt{e}} \)
• B. \( \frac{3}{3 - \sqrt{e}} \)
• C. \( \frac{2}{1 + \sqrt{e}} \)
• D. \( \frac{1}{2 - \sqrt{e}} \)

Answer 1

The Correct Option is D

We are given the differential equation:

\[\frac{dy}{dx} = 2x(x + y)^3 - x(x + y) - 1\]

Let \( x + y = t \). Therefore, we have:

\[\frac{dt}{dx} = 2xt^3 - xt - 1\]

This simplifies to:

\[\frac{dt}{dx} = t^2 \quad \text{and} \quad \frac{dt}{dx} = x^2 \text{ for } x = 0\]

Now solve the equation:

\[\int \frac{dz}{2(2z - z)} = \int x dx\]

After solving:

\[\ln \left( \frac{z - 1}{z} \right) = x^2 + k\]

Thus, \( z = \frac{1}{2 - \sqrt{e}} \).