Question

The integrating factor of $\frac{dy}{dx} + 3x = 2y$ is

• A. $e^{3x}$
• B. $e^{-2x}$
• C. $e^x$
• D. $0$

Hint:

Always ensure the coefficient of $\frac{dy}{dx}$ is 1 and all terms involving '$y$' are on the left side before identifying $P(x)$. A common mistake is forgetting the negative sign when $P(x)$ is negative.

Answer 1

The Correct Option is B

Step 1: Standard Linear Form: The standard form of a first-order linear differential equation is: $$\frac{dy}{dx} + P(x)y = Q(x)$$ Rearranging the given equation $\frac{dy}{dx} + 3x = 2y$: $$\frac{dy}{dx} - 2y = -3x$$

Step 2: Identify $P(x)$: By comparing with the standard form, we find: $$P(x) = -2$$

Step 3: Calculate the Integrating Factor (IF): The formula for the integrating factor is $IF = e^{\int P(x) dx}$. Substituting $P(x) = -2$: $$IF = e^{\int -2 \, dx}$$ $$IF = e^{-2x}$$ Thus, the integrating factor required to solve the equation is $e^{-2x}$.