Question

Calculate \( y(\log 2) \) if \( \dfrac{dy}{dx} = y + 5 \) and \( y(0) = 4 \).

• A. \(12\)
• B. \(13\)
• C. \(14\)
• D. \(15\)

Hint:

Many differential equations of the form \( \frac{dy}{dx}=y+c \) can be solved quickly by separating variables and integrating logarithmically.

Answer 1

The Correct Option is B

Concept: A first-order linear differential equation of the form \[ \frac{dy}{dx} = y + c \] can be solved using separation of variables.

Step 1: Rewrite the differential equation. \[ \frac{dy}{dx} = y + 5 \] \[ \frac{dy}{y+5} = dx \]

Step 2: Integrate both sides. \[ \int \frac{1}{y+5}dy = \int dx \] \[ \log|y+5| = x + C \]

Step 3: Apply the initial condition \(y(0)=4\). \[ \log(4+5) = C \] \[ C = \log 9 \] Thus, \[ \log(y+5) = x + \log 9 \] \[ y+5 = 9e^{x} \]

Step 4: Substitute \(x=\log 2\). \[ y+5 = 9e^{\log 2} \] \[ y+5 = 18 \] \[ y = 13 \]