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

Which of the following is the general solution to the differential equation $y^{''} + y = 5 x \sin (x)$ using the method of undetermined coefficients?

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

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

y = C_{1} \cos (x) + C_{2} \sin (x) + x \cos (x) + 2 x \sin (x) - 5 x \cos (x)

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

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Verified step by step guidance

Step 1: Recognize that the differential equation y'' + y = 5x sin(x) is a second-order non-homogeneous differential equation. The solution consists of two parts: the complementary solution (associated with the homogeneous equation y'' + y = 0) and the particular solution (associated with the non-homogeneous term 5x sin(x)).

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

Step 3: Use the method of undetermined coefficients to find the particular solution y_p for the non-homogeneous term 5x sin(x). Since the right-hand side involves x sin(x), propose a trial solution of the form y_p = (Ax + B) sin(x) + (Cx + D) cos(x), where A, B, C, and D are coefficients to be determined.

Step 4: Substitute the trial solution y_p into the original differential equation y'' + y = 5x sin(x). Compute y_p', y_p'' and substitute these into the equation. Collect terms involving sin(x) and cos(x), and equate coefficients of like terms to solve for A, B, C, and D.

Step 5: Combine the complementary solution y_c and the particular solution y_p to form the general solution: y = y_c + y_p. Simplify the expression to match one of the given options, ensuring all coefficients are correctly determined.