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: Substitute:
Let $z=y+5$. Then $\frac{dz}{dx}=\frac{dy}{dx}=z$.

Step 2: Solve for z:
$z$ grows exponentially: $z=z_0e^{x}$. At $x=0$, $z_0=4+5=9$.

Step 3: Evaluate:
At $x=\log2$, $z=9e^{\log2}=18$. Then $y=z-5=13$.

Step 4: Sanity check:
The derivative at that point is $y+5=18$, which equals $9e^{x}$ at $x=\log 2$. Good.

Final Answer:
We get $y(\log2)=13$ (option D). \[ \boxed{13} \]