Math Problem Statement

Which one of the following is the integrating factor of dy/dx + (1/x)y = x^2?

Solution

The question asks to find the integrating factor of the given first-order linear differential equation:

dydx+1xy=x2\frac{dy}{dx} + \frac{1}{x}y = x^2

General Form of a First-Order Linear Differential Equation:

A first-order linear differential equation has the general form: dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x) where P(x)P(x) and Q(x)Q(x) are functions of xx.

Integrating Factor:

The integrating factor (μ(x)\mu(x)) is given by: μ(x)=eP(x)dx\mu(x) = e^{\int P(x) \, dx} where P(x)P(x) is the coefficient of yy in the equation.

In this case, P(x)=1xP(x) = \frac{1}{x}.

Calculation of Integrating Factor:

μ(x)=e1xdx=elnx=x\mu(x) = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = x

Thus, the integrating factor is xx.

Answer:

The correct option is:

B) xx.

Would you like more details or have any questions?

Related Questions:

  1. How do you solve a differential equation after finding the integrating factor?
  2. Can you explain how the integrating factor method works in general?
  3. What happens if P(x)P(x) is not 1x\frac{1}{x}? How does that change the integrating factor?
  4. What if the differential equation were non-linear? How would that affect solving it?
  5. How can you verify that the integrating factor you found is correct?

Tip:

When solving a linear differential equation, always write it in the standard form first to easily identify P(x)P(x) and apply the integrating factor formula correctly.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Integrating Factors
Linear Equations

Formulas

Integrating Factor: μ(x) = e^(∫P(x) dx)

Theorems

Integrating Factor Method for First-Order Linear Differential Equations

Suitable Grade Level

Grades 11-12