If $\frac{dy}{dx} = y + 5$ and $y(0) = 4$, then $y(\log 2)$ is equal to
Show 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.
Step 1: Understanding the Question:
This is a first-order ordinary differential equation with an initial condition. We need to solve for $y$ and evaluate it at a specific $x$. Step 2: Key Formula or Approach:
Use the variable separable method: $\int \frac{dy}{g(y)} = \int f(x) dx$. Step 3: Detailed Explanation:
Given: $\frac{dy}{dx} = y + 5$
Separating variables:
\[ \frac{dy}{y + 5} = dx \]
Integrating both sides:
\[ \int \frac{dy}{y + 5} = \int dx \]
\[ \log(y + 5) = x + C \]
Apply initial condition $y(0) = 4$:
\[ \log(4 + 5) = 0 + C \implies C = \log 9 \]
The general solution becomes:
\[ \log(y + 5) = x + \log 9 \]
Now, find $y$ when $x = \log 2$:
\[ \log(y + 5) = \log 2 + \log 9 \]
\[ \log(y + 5) = \log(2 \times 9) \]
\[ \log(y + 5) = \log 18 \]
\[ y + 5 = 18 \implies y = 13 \] Step 4: Final Answer:
The value of $y(\log 2)$ is $13$.