Step 1: Understanding the Question:
This is a first-order ordinary differential equation. We need to find the specific solution using the initial condition and then evaluate it at a specific point.
Step 2: Key Formula or Approach:
We use the method of Separation of Variables:
\[ \int \frac{dy}{g(y)} = \int f(x) dx \]
Step 3: Detailed Explanation:
Separating the variables:
\[ \frac{dy}{y+5} = dx \]
Integrating both sides:
\[ \int \frac{1}{y+5} dy = \int dx \]
\[ \log|y+5| = x + C \]
Using the initial condition \(y(0) = 4\):
\[ \log|4+5| = 0 + C \Rightarrow C = \log 9 \]
The general equation becomes:
\[ \log|y+5| = x + \log 9 \]
Taking the exponential of both sides:
\[ y+5 = 9e^x \]
To find \(y(\log 2)\), substitute \(x = \log 2\):
\[ y+5 = 9e^{\log 2} \]
Since \(e^{\log 2} = 2\):
\[ y+5 = 9(2) = 18 \]
\[ y = 18 - 5 = 13 \]
Step 4: Final Answer:
The value of \(y\) at \(x = \log 2\) is 13.