Question

If the function \(y=f(x)\) satisfies the differential equation \(x\frac{dy}{dx}=xy+y\) and \(y(1)=e\), then which of the following is true?

• A. \(f(x)\) is not continuous at \(x=0\)
• B. \(f(x)\) is decreasing in \((-1,\infty )\)
• C. \(f(x)\) is increasing in \((-1,\infty )\)
• D. \(f(x)\) has maxima at \(x=1\)

Hint:

Separate the variables, find y with y(1) = e, then check where it increases.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Separate the variables: \(\dfrac{dy}{y}=\left(1+\dfrac{1}{x}\right)dx\).

Step 2: Integrate
\(\log y=x+\log x+C\), so \(y=Kxe^{x}\).
Using \(y(1)=e\) gives \(K=1\), so \(f(x)=xe^{x}\).

Step 3: Differentiate
\(f'(x)=e^{x}(x+1)\).

Step 4: Compare the options
\(f'(x)>0\) for \(x>-1\), so \(f\) is increasing in \((-1,\infty )\).
Also \(f(0)=0\) is continuous, and there is no maximum at \(x=1\).

Final Answer:
The function is increasing in \((-1,\infty )\). \[ \boxed{\text{(C) }\text{f(x) is increasing in }(-1,\infty )} \]