<p>Let \(y = f(x)\) satisfy the differential equation \(xy(1 + y)dx = dy\). If \(f(0) = 1\) and \(f(2) = \frac{e^2}{k - e^2}\), then find the value of k.</p>
Step-by-Step Solution
Key Concept: Solve the separable differential equation using partial fractions and apply initial conditions to find the parameter.
<p><strong>Step 1:</strong> Rewrite the differential equation as: $\frac{dy}{dx} = xy(1+y)$</p><p><strong>Step 2:</strong> Separate variables: $\frac{dy}{y(1+y)} = x dx$</p><p><strong>Step 3:</strong> Use partial fractions: $\frac{1}{y(1+y)} = \frac{1}{y} - \frac{1}{1+y}$</p><p><strong>Step 4:</strong> Integrate: $\ln|y| - \ln|1+y| = \frac{x^2}{2} + C$</p><p><strong>Step 5:</strong> Apply initial condition $f(0) = 1$: $\ln 1 - \ln 2 = C$, so $C = -\ln 2$</p><p><strong>Step 6:</strong> At $x = 2$: $\ln\left|\frac{y}{1+y}\right| = 2 - \ln 2$</p><p><strong>Step 7:</strong> Solve for y and match with given form to find $k = 2$</p>
Correct Answer: 2