Question

If \(\phi(x)\) is a differentiable function, then the solution of the differential equation \[ dy+y\phi'(x)-\phi(x)\phi'(x)\,dx=0 \] is

• A. \(y=\{\phi(x)-1\}+Ce^{-\phi(x)}\)
• B. \(y\phi(x)=\{\phi(x)\}^2+C\)
• C. \(ye^{\phi(x)}=\phi(x)e^{\phi(x)}+C\)
• D. \(y-\phi(x)=\phi(x)e^{-\phi(x)}\)

Hint:

Linear DE: multiply by integrating factor.

Answer 1

The Correct Option is C

Step 1: Rearrange: \[ \frac{dy}{dx}+\phi'(x)y=\phi(x)\phi'(x) \]
Step 2: This is a linear differential equation.
Step 3: Integrating factor: \[ IF=e^{\int\phi'(x)dx}=e^{\phi(x)} \]
Step 4: \[ \frac{d}{dx}\big(ye^{\phi(x)}\big)=\phi(x)\phi'(x)e^{\phi(x)} \]
Step 5: Integrating: \[ ye^{\phi(x)}=\phi(x)e^{\phi(x)}+C \]