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 $y^{''} + 2 y^{'} = 2 x + 7 - e^{- 2 x}$ using the method of undetermined coefficients. Which of the following is the general solution?

y (x) = C_{1} + C_{2} e^{- 2 x} + x^{2} + 7 x + e^{- 2 x}

y (x) = C_{1} + C_{2} e^{- 2 x} + x^{2} + 7 x - \frac{1}{2} e^{- 2 x}

y (x) = C_{1} + C_{2} e^{- 2 x} + x^{2} + 7 x

y (x) = C_{1} e^{- 2 x} + C_{2} + 2 x + 7 - e^{- 2 x}

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Verified step by step guidance

Step 1: Identify the type of differential equation. The given equation y'' + 2y' = 2x + 7 - e^{-2x} is a second-order linear non-homogeneous differential equation. The method of undetermined coefficients is suitable for solving this type of equation.

Step 2: Solve the complementary (homogeneous) equation. The complementary equation is y'' + 2y' = 0. Solve this by finding the characteristic equation, which is r^2 + 2r = 0. Factorize to get r(r + 2) = 0, giving roots r = 0 and r = -2. Thus, the complementary solution is y_c(x) = C_1 + C_2 e^{-2x}, where C_1 and C_2 are constants.

Step 3: Determine the form of the particular solution. The non-homogeneous term on the right-hand side is 2x + 7 - e^{-2x}. Break this into two parts: a polynomial (2x + 7) and an exponential term (-e^{-2x}). For the polynomial, assume a particular solution of the form Ax^2 + Bx + C. For the exponential term, since e^{-2x} is already part of the complementary solution, multiply it by x to avoid duplication, resulting in a term of the form Dx e^{-2x}. Combine these to form the particular solution: y_p(x) = Ax^2 + Bx + C + Dx e^{-2x}.

Step 4: Substitute y_p(x) into the original equation. Compute y_p'(x) and y_p''(x), then substitute these along with y_p(x) into the differential equation y'' + 2y' = 2x + 7 - e^{-2x}. Collect like terms and equate coefficients of x^2, x, constant terms, and e^{-2x} to solve for A, B, C, and D.

Step 5: Combine the complementary and particular solutions. Once the coefficients A, B, C, and D are determined, the general solution is y(x) = y_c(x) + y_p(x). Substitute the values of A, B, C, and D into y_p(x) and add it to y_c(x) = C_1 + C_2 e^{-2x} to obtain the final general solution.