Differential Equations Question - None

Question

If $\dfrac{dy}{dx} = y + 3 > 0$ and $y(0) = 2$, then $y(\ln 2)$ is equal to

Accepted Answer

Step 1: Separate variables

Given: dy/dx = y + 3 with y(0) = 2

Rearranging: dy/(y + 3) = dx

Step 2: Integrate both sides

∫ dy/(y + 3) = ∫ dx

ln|y + 3| = x + C

Step 3: Apply initial condition y(0) = 2

ln|2 + 3| = 0 + C

ln(5) = C

Step 4: Write the particular solution

ln|y + 3| = x + ln(5)

|y + 3| = e^(x + ln 5) = 5e^x

Since y(0) = 2 > -3, we have y + 3 > 0, so:

y + 3 = 5e^x

y = 5e^x - 3

Step 5: Evaluate at x = ln(2)

y(ln 2) = 5e^(ln 2) - 3 = 5(2) - 3 = 10 - 3 = 7

∴ Answer: 7

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