Question

The integrating factor of the differential equation $(1 + x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0$ is}

• A. $\log(1 - x)$
• B. $\log(1 + x^2)$
• C. $1 + x^2$
• D. $1 + 2x$

Hint:

Notice that the left-hand side of the original equation $(1+x^2)\frac{dy}{dx} + 2xy$ is already the exact derivative of $(1+x^2)y$ by the product rule.
This immediately shows that the integrating factor must be $1+x^2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question asks for the integrating factor of a first-order linear ordinary differential equation.
The integrating factor is a function used to transform a non-exact differential equation into an exact one.

Step 2: Key Formula or Approach: A first-order linear differential equation has the standard form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \] The integrating factor ($I.F.$) is calculated as:
\[ I.F. = e^{\int P(x) dx} \]

Step 3: Detailed Explanation:

• Rewrite the given differential equation in standard form:
\[ (1 + x^2)\frac{dy}{dx} + 2xy = 4x^2 \]
• Divide the entire equation by $(1 + x^2)$:
\[ \frac{dy}{dx} + \left(\frac{2x}{1 + x^2}\right)y = \frac{4x^2}{1 + x^2} \]
• Identify $P(x)$ by comparing it with the standard linear form:
\[ P(x) = \frac{2x}{1 + x^2} \]
• Integrate $P(x)$ with respect to $x$:
\[ \int P(x) dx = \int \frac{2x}{1 + x^2} dx \]
• Let $u = 1 + x^2$, which gives $du = 2x dx$. The integral becomes:
\[ \int \frac{du}{u} = \ln|u| = \ln(1 + x^2) \]
• Calculate the integrating factor:
\[ I.F. = e^{\int P(x) dx} = e^{\ln(1 + x^2)} \]
• Since $e^{\ln(f(x))} = f(x)$, we get:
\[ I.F. = 1 + x^2 \]

Step 4: Final Answer:
The integrating factor is $1 + x^2$, which corresponds to option (C).