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

Given the linear system of differential equations $\begin{array}{l} \frac{\partial x}{\partial t} = 2 x + y \\ \frac{\partial y}{\partial t} = x + 2 y \end{array}$ find the most general real-valued solution.

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

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

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

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

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

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

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

y (t) = C_{1} e^{2 t} - C_{2} e^{- 2 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 $ \frac{d}{dt} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $. This allows us to analyze the system using linear algebra techniques.

Step 2: Find the eigenvalues of the coefficient matrix $ \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} $. To do this, solve the characteristic equation $ \det(A - \lambda I) = 0 $, where $ A $ is the matrix and $ \lambda $ represents the eigenvalues. The determinant is $ \det \begin{bmatrix} 2 - \lambda & 1 \\ 1 & 2 - \lambda \end{bmatrix} = (2 - \lambda)^2 - 1 $.

Step 3: Simplify the characteristic equation $ (2 - \lambda)^2 - 1 = 0 $ to find the eigenvalues. Expand and solve for $ \lambda $: $ \lambda^2 - 4\lambda + 3 = 0 $. Factorize the quadratic equation to obtain $ \lambda = 3 $ and $ \lambda = 1 $. These are the eigenvalues of the matrix.

Step 4: For each eigenvalue, find the corresponding eigenvector by solving $ (A - \lambda I) \vec{v} = 0 $. For $ \lambda = 3 $, solve $ \begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = 0 $. For $ \lambda = 1 $, solve $ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = 0 $. Normalize the eigenvectors if necessary.

Step 5: Construct the general solution using the eigenvalues and eigenvectors. The solution takes the form $ \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = C_1 e^{\lambda_1 t} \vec{v}_1 + C_2 e^{\lambda_2 t} \vec{v}_2 $, where $ \lambda_1 $ and $ \lambda_2 $ are the eigenvalues, and $ \vec{v}_1 $ and $ \vec{v}_2 $ are the corresponding eigenvectors. Substitute the values of $ \lambda $ and $ \vec{v} $ to express $ x(t) $ and $ y(t) $ explicitly.