Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

13. Intro to Differential Equations

Basics of Differential Equations

Calculus

13. Intro to Differential Equations

Basics of Differential Equations

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Multiple Choice

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

x^{- 1}

\sin (x)

e^{x}

x

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Verified step by step guidance

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).

Step 2: Identify the coefficient of y in the standard form, which is -1/x. This coefficient is used to determine the integrating factor.

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.

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).

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.