Question

If \(\frac{dy}{dx} = y+5\) and \(y(0) = 4\) then \(y(log2)\) is equal to

• A. \(2\)
• B. \(5\)
• C. \(7\)
• D. \(13\)

Hint:

Separate variables: dy/(y+5) = dx, integrate, then use y(0)=4.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The equation is first order with separable variables. We move all \(y\) terms to one side and all \(x\) terms to the other, then integrate.

Step 2: Separate and integrate:
\[ \frac{dy}{y+5}=dx \Rightarrow \log|y+5|=x+c \]
So \(y+5=Ce^{x}\).

Step 3: Use the initial condition:
At \(x=0\), \(y=4\), so \(9=C\). Hence \(y=9e^{x}-5\).

Step 4: Evaluate at x = log 2:
Since \(e^{\log 2}=2\), we get \(y(\log2)=9(2)-5=13\).

Step 5: Why the other options are wrong:
Option (A) 2 and option (B) 5 come from dropping the constant or the factor \(9\). Option (C) 7 comes from forgetting to subtract or add the shift of 5 correctly. Only \(13\) follows from \(C=9\) and \(y=9e^x-5\).

Final Answer:
The value is \(13\), option (D). \[ \boxed{13} \]