Question

Given that \[ \frac{dy}{dx}=ye^x \] and when \[ x=0,\quad y=e, \] then the value of \(y\) (\(y>0\)) when \(x=1\) is

• A. \[ \frac1e \]
• B. \[ e \]
• C. \[ e^e \]
• D. \[ \log e \] \bigskip

Hint:

For equations of the form \[ \frac{dy}{dx}=y\,f(x), \] separate variables: \[ \frac{dy}{y}=f(x)\,dx, \] integrate, and then apply the initial condition.

Answer 1

The Correct Option is C

Concept: Separable ODE. Separate, integrate, apply initial condition, evaluate at \(x=1\).

Step 1: \(dy/dx = y e^x \Rightarrow dy/y = e^x dx\). Integrate: \(\ln y = e^x + C \Rightarrow y = Ke^{e^x}\).

Step 2: \(x=0, y=e\): \(e = Ke^{e^0} = Ke \Rightarrow K=1\). So \(y = e^{e^x}\). At \(x=1\): \(y = e^e\).

Step 3: Write the final answer. \(\boxed{e^e}\)