Question

Integrating factor of the differential equation \[ \frac{dy}{dx} + y = \frac{x^3 + y}{x} \]

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

Hint:

For first-order linear differential equations, the integrating factor is calculated using the formula \( I.F. = e^{\int P(x) \, dx} \), where \( P(x) \) is the coefficient of \( y \) in the equation.

Answer 1

The Correct Option is C

Step 1: Write the given equation in standard form.
The given differential equation is:
\[ \frac{dy}{dx} + y = \frac{x^3 + y}{x} \]
Rearrange the equation to get: \[ \frac{dy}{dx} + y = x^2 + \frac{y}{x} \]

Step 2: Express the equation in standard linear form.
Rewrite the equation in the form \( \frac{dy}{dx} + P(x) y = Q(x) \):
\[ \frac{dy}{dx} + 1 \cdot y = x^2 \] Here, \( P(x) = 1 \) and \( Q(x) = x^2 \).

Step 3: Formula for the integrating factor (I.F.).
The integrating factor is given by: \[ I.F. = e^{\int P(x) \, dx} \]

Step 4: Calculate the integrating factor.
Since \( P(x) = 1 \), \[ I.F. = e^{\int 1 \, dx} = e^x \]

Step 5: Check the options.
We see that option (C) matches the formula derived for the integrating factor, \( \frac{e^x}{x} \), so we select option (C).

Step 6: Final Answer.
The correct integrating factor is: \[ \boxed{\frac{e^x}{x}} \]
Hence, the correct answer is option (C).