Consider the following differential equation: $x\frac{dy}{dx}-y=x^2\sin \left(x\right)$. Which of the following is the correct integrating factor to solve this equation?

• A. $x^{-1}$
• B. $\sin \left(x\right)$
• C. $e^x$
• D. $x$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the given differential equation is in the form of a first-order linear differential equation: x * (dy/dx) - y = x^2 * sin(x). Rewrite it in standard form by dividing through by x (assuming x ≠ 0): (dy/dx) - (y/x) = x * sin(x).
2. Step 2: Identify the coefficient of y in the standard form, which is -1/x. This coefficient is used to determine the integrating factor.
3. Step 3: Recall the formula for the integrating factor in a first-order linear differential equation: IF = e^(∫P(x) dx), where P(x) is the coefficient of y. Here, P(x) = -1/x.
4. Step 4: Compute the integrating factor: IF = e^(∫(-1/x) dx). The integral of -1/x is -ln(x), so the integrating factor becomes e^(-ln(x)). Simplify this expression using the property of exponents: e^(-ln(x)) = x^(-1).
5. Step 5: Conclude that the correct integrating factor for this differential equation is x^(-1). This integrating factor will be used to solve the equation by multiplying through by it and proceeding with the solution process.