Question

Let a curve \( y = f(x) \) pass through the points \( (0,5) \) and \( (\log 2, k) \). If the curve satisfies the differential equation: \[ 2(3+y)e^{2x}dx - (7+e^{2x})dy = 0, \] then \( k \) is equal to:

• A. \( 16 \)
• B. \( 8 \)
• C. \( 32 \)
• D. \( 4 \)

Hint:

When solving a differential equation, always check whether it's separable or requires an integrating factor. Substituting boundary conditions correctly helps determine constants.

Answer 1

The Correct Option is B

Step 1: Expressing the differential equation. Rewriting the given equation: \[ \frac{dy}{dx} = \frac{2(3+y)e^{2x}}{7+e^{2x}}. \] Separating variables: \[ \frac{dy}{(3+y)} = \frac{2e^{2x}dx}{7+e^{2x}}. \] Step 2: Finding the integrating factor (I.F.). \[ I.F. = e^{-\int \frac{2e^{2x}dx}{7+e^{2x}}}. \] Using substitution, \[ I.F. = \frac{1}{7 + e^{2x}}. \] Step 3: Solving for \( y \). Multiplying by the integrating factor: \[ y \cdot \frac{1}{7+e^{2x}} = \int \frac{6e^{2x}dx}{(7+e^{2x})^2}. \] Integrating, we get: \[ \frac{y}{7+e^{2x}} = \frac{-3}{7+e^{2x}} + C. \] Step 4: Applying the initial condition \( (0,5) \). \[ \frac{5}{8} = \frac{-3}{8} + C. \] Solving for \( C \), \[ C = 1. \] Step 5: Finding \( k \) at \( x = \log 2 \). \[ y = -3 + 7 + e^{2x} = e^{2x} + 4. \] \[ k = e^{2 \log 2} + 4 = 4 + 4 = 8. \]