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 $5 y ″ - 10 y' + 10 y = e^{x} s e c x$ using the method of variation of parameters. Which of the following is the general solution?

y (x) = C_{1} e^{x} + C_{2} e^{2 x} + e^{x} c o s x

y (x) = C_{1} e^{x} + C_{2} e^{2 x} + \frac{1}{5} e^{x} s e c x

y (x) = C_{1} e^{x} + C_{2} e^{2 x} + e^{x} s i n x

y (x) = C_{1} e^{x} + C_{2} e^{2 x} + \frac{1}{5} e^{x} t a n x

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Verified step by step guidance

Step 1: Start by solving the corresponding homogeneous equation. The given differential equation is 5y'' - 10y' + 10y = e^x sec(x). For the homogeneous part, set the right-hand side to 0: 5y'' - 10y' + 10y = 0. Solve this characteristic equation: 5r^2 - 10r + 10 = 0.

Step 2: Solve the characteristic equation 5r^2 - 10r + 10 = 0. Divide through by 5 to simplify: r^2 - 2r + 2 = 0. Use the quadratic formula r = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = -2, and c = 2. Compute the discriminant b^2 - 4ac = (-2)^2 - 4(1)(2) = -4, which is negative. This indicates complex roots.

Step 3: The roots of the characteristic equation are r = 1 ± i. The general solution to the homogeneous equation is then y_h(x) = e^x(C_1 cos(x) + C_2 sin(x)), where C_1 and C_2 are constants determined by initial conditions.

Step 4: To solve the non-homogeneous equation using the method of variation of parameters, assume a particular solution of the form y_p(x) = u_1(x)e^x cos(x) + u_2(x)e^x sin(x), where u_1(x) and u_2(x) are functions to be determined. Substitute y_p(x) into the original equation and use the Wronskian of the solutions e^x cos(x) and e^x sin(x) to find u_1'(x) and u_2'(x).

Step 5: Compute u_1'(x) and u_2'(x) using the formulas u_1'(x) = -y_2(x)g(x)/W(x) and u_2'(x) = y_1(x)g(x)/W(x), where y_1(x) = e^x cos(x), y_2(x) = e^x sin(x), g(x) = e^x sec(x), and W(x) is the Wronskian of y_1 and y_2. Integrate u_1'(x) and u_2'(x) to find u_1(x) and u_2(x), then substitute back into y_p(x). Combine y_h(x) and y_p(x) to form the general solution.