Question

If $\frac{dy}{dx} = y + 5$ and $y(0) = 4$, then $y(\log 2)$ is equal to

• A. $13$
• B. $15$
• C. $18$
• D. $9$

Hint:

While solving differential equations involving logarithms, writing the constant as \(\log C\) often simplifies calculations because logarithmic identities like \(\log a + \log b = \log(ab)\) can be applied directly.

Answer 1

The Correct Option is A

Concept:
The given equation is a first–order differential equation. It can be solved using the separation of variables method: \[ \int \frac{dy}{g(y)} = \int f(x)\,dx \]
Step 1: {Separate the variables.} Given: \[ \frac{dy}{dx} = y + 5 \] Rearranging: \[ \frac{dy}{y+5} = dx \]
Step 2: {Integrate both sides.} \[ \int \frac{dy}{y+5} = \int dx \] \[ \log|y+5| = x + C \]
Step 3: {Apply the initial condition \(y(0)=4\).} Substitute \(x=0\), \(y=4\): \[ \log(4+5) = C \] \[ C = \log 9 \] Thus the solution becomes: \[ \log(y+5) = x + \log 9 \]
Step 4: {Find \(y\) when \(x=\log 2\).} \[ \log(y+5) = \log 2 + \log 9 \] Using logarithmic property: \[ \log(y+5) = \log(18) \] \[ y+5 = 18 \] \[ y = 13 \]
Step 5: {Conclusion.} \[ y(\log 2) = 13 \]