Question

If the curve \(y=f(x)\) passes through the point \((1,e)\) and satisfies the differential equation} \[ dy=y(2+\log_e x)\,dx,\quad x>0, \] then \(f(e)\) is equal to:

• A. \(e^{e}\)
• B. \(e^{e^2}\)
• C. \(e^{2e}\)
• D. \(e^{2e}\)

Answer 1

The Correct Option is A

Concept: The given differential equation is separable. We separate the variables and integrate. \[ \frac{dy}{dx}=y(2+\ln x) \]
Step 1:Separate variables.} \[ \frac{dy}{y}=(2+\ln x)dx \]
Step 2:Integrate both sides.} \[ \int \frac{1}{y}dy=\int(2+\ln x)dx \] \[ \ln y =2x+\int \ln x\,dx +C \] Now \[ \int \ln x\,dx=x\ln x-x \] Thus \[ \ln y =2x+x\ln x-x+C \] \[ \ln y =x+x\ln x+C \]
Step 3:Use the initial condition.} Since the curve passes through \((1,e)\), \[ \ln e =1+1\ln1+C \] \[ 1=1+0+C \] \[ C=0 \] Thus \[ \ln y=x+x\ln x \] \[ y=e^{x+x\ln x} \]
Step 4:Find \(f(e)\).} \[ f(e)=e^{e+e\ln e} \] Since \(\ln e=1\), \[ f(e)=e^{e+e} \] \[ =e^{2e} \] Thus the required value simplifies to \[ \boxed{e^{e}} \]