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

Solve the differential equation by variation of parameters: $y^{″} + y = \sin ⁠ 2 x$ . What is the general solution $y (x)$ ?

y (x) = C_{1} e^{x} + C_{2} e^{- x} + \frac{1}{2} \sin ⁠ 2 x

y (x) = C_{1} \cos ⁠ x + C_{2} \sin ⁠ x + \frac{1}{2} \sin ⁠ 2 x

y (x) = C_{1} \cos ⁠ x + C_{2} \sin ⁠ x + \frac{1}{3} \cos ⁠ 2 x

y (x) = C_{1} \cos ⁠ x + C_{2} \sin ⁠ x - \frac{1}{3} \sin ⁠ 2 x

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Verified step by step guidance

Step 1: Identify the type of differential equation. The given equation y'' + y = sin(2x) is a second-order linear non-homogeneous differential equation. The method of variation of parameters is suitable for solving this type of equation.

Step 2: Solve the corresponding homogeneous equation y'' + y = 0. The characteristic equation is r^2 + 1 = 0, which has roots r = ±i. Therefore, the general solution to the homogeneous equation is y_h(x) = C_1 cos(x) + C_2 sin(x), where C_1 and C_2 are constants.

Step 3: Use variation of parameters to find a particular solution y_p(x) for the non-homogeneous equation. Assume y_p(x) = u_1(x) cos(x) + u_2(x) sin(x), where u_1(x) and u_2(x) are functions to be determined.

Step 4: Compute u_1(x) and u_2(x) using the formulas derived from variation of parameters. These involve integrating expressions that depend on the Wronskian of the solutions to the homogeneous equation and the non-homogeneous term sin(2x). Specifically, u_1'(x) = -sin(x) * sin(2x) / W(x) and u_2'(x) = cos(x) * sin(2x) / W(x), where W(x) = cos(x) * sin(x) - sin(x) * cos(x) = 1.

Step 5: Combine the homogeneous solution y_h(x) and the particular solution y_p(x) to form the general solution y(x). After performing the necessary integrations and simplifications, the general solution is y(x) = C_1 cos(x) + C_2 sin(x) - (1/3) sin(2x).