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 $4 y^{″} + y^{'} + y = x^{2} - 5 x$ using the method of undetermined coefficients. What is the general solution?

y = C_{1} e^{- \frac{1}{8} x} + C_{2} e^{- \frac{1}{2} x} + x^{2} - 5 x + 10

y = C_{1} e^{- \frac{1}{8} x} + C_{2} e^{- \frac{1}{2} x} + x^{2} - 5 x

y = C_{1} e^{- \frac{1}{8} x} + C_{2} e^{- \frac{1}{2} x} + x^{2} - 5 x + 10

y = C_{1} e^{- \frac{1}{8} x} + C_{2} e^{- \frac{1}{2} x} + x^{2} - 5 x + 20

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Verified step by step guidance

Step 1: Identify the type of differential equation. The given equation is a second-order linear differential equation with constant coefficients: 4y'' + y' + y = x^2 - 5x. The right-hand side (x^2 - 5x) suggests that the method of undetermined coefficients is appropriate for solving this equation.

Step 2: Solve the homogeneous equation. First, consider the associated homogeneous equation: 4y'' + y' + y = 0. Solve this by finding the characteristic equation, which is obtained by substituting y = e^(rx): 4r^2 + r + 1 = 0. Solve this quadratic equation for r to find the roots, which will determine the complementary solution.

Step 3: Determine the particular solution. For the non-homogeneous part (x^2 - 5x), propose a trial solution based on the form of the right-hand side. Since the right-hand side is a polynomial of degree 2, the trial solution should be of the form y_p = Ax^2 + Bx + C. Substitute y_p into the original equation to solve for the coefficients A, B, and C.

Step 4: Combine the solutions. The general solution to the differential equation is the sum of the complementary solution (from Step 2) and the particular solution (from Step 3). Write the general solution as y = y_c + y_p, where y_c is the solution to the homogeneous equation and y_p is the particular solution.

Step 5: Verify the solution. Substitute the general solution back into the original differential equation to ensure that it satisfies the equation. This step confirms the correctness of the solution and ensures that all terms are accounted for.