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

2. Intro to Derivatives

Differentiability

Calculus

2. Intro to Derivatives

Differentiability

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Multiple Choice

Given the system of differential equations $\frac{d x}{d t} = 2 x + 3 y$ and $\frac{d y}{d t} = 6 x + 5 y$ , which of the following is the general solution for $x (t)$ and $y (t)$ ?

x (t) = C_{1} e^{5 t} + C_{2} e^{2 t}

y (t) = 6 C_{1} e^{5 t} + 3 C_{2} e^{2 t}

x (t) = C_{1} e^{t} + C_{2} e^{- 8 t}

y (t) = 3 C_{1} e^{t} + 2 C_{2} e^{- 8 t}

x (t) = C_{1} e^{8 t} + C_{2} e^{- t}

y (t) = 2 C_{1} e^{8 t} - C_{2} e^{- t}

x (t) = C_{1} e^{3 t} + C_{2} e^{- 3 t}

y (t) = C_{1} e^{3 t} - C_{2} e^{- 3 t}

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Verified step by step guidance

Step 1: Start by writing the system of differential equations in matrix form. Represent the system as d/dt [x, y]^T = A [x, y]^T, where A is the coefficient matrix. Here, A = [[2, 3], [6, 5]].

Step 2: Find the eigenvalues of matrix A by solving the characteristic equation det(A - λI) = 0, where λ represents the eigenvalues and I is the identity matrix. This involves calculating det([[2-λ, 3], [6, 5-λ]]) = 0.

Step 3: Once the eigenvalues are determined, find the corresponding eigenvectors for each eigenvalue by solving (A - λI)v = 0, where v is the eigenvector. This involves substituting each eigenvalue into the matrix equation and solving for v.

Step 4: Use the eigenvalues and eigenvectors to construct the general solution for x(t) and y(t). The solution takes the form x(t) = C_1 e^(λ1t) + C_2 e^(λ2t) and y(t) = linear combinations of the eigenvectors scaled by e^(λ1t) and e^(λ2t).

Step 5: Compare the derived general solution with the given options to identify the correct answer. Ensure the coefficients and exponential terms match the eigenvalues and eigenvectors obtained from the matrix A.