Solve the differential equation using variation of parameters: y''+3y'+2y=13+ex. Which of the following is the general solution?

y = C 1 e - x + C 2 e - 2 x + 1 6 + 1 2 x e x

Solve the differential equation using variation of parameters: y '' + 3 y ' + 2 y = 1 3 + e x . Which of the following is the general solution?

y = C 1 e - x + C 2 e - 2 x + 1 6 + 1 2 x e x

y = C 1 e - x + C 2 e - 2 x + 1 6 + 1 3 e x

y = C 1 e - x + C 2 e - 2 x + 1 3 + 1 2 x e x

y = C 1 e - x + C 2 e - 2 x + 1 3 + 1 3 e x

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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
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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 using variation of parameters: $y^{''} + 3 y^{'} + 2 y = \frac{1}{3} + e^{x}$ . Which of the following is the general solution?

y = C_{1} e^{- x} + C_{2} e^{- 2 x} + \frac{1}{6} + \frac{1}{3} e^{x}

y = C_{1} e^{- x} + C_{2} e^{- 2 x} + \frac{1}{3} + \frac{1}{2} x e^{x}

y = C_{1} e^{- x} + C_{2} e^{- 2 x} + \frac{1}{6} + \frac{1}{2} x e^{x}

y = C_{1} e^{- x} + C_{2} e^{- 2 x} + \frac{1}{3} + \frac{1}{3} e^{x}

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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 non-homogeneous differential equation: y'' + 3y' + 2y = 1/3 + e^x.

Step 2: Solve the corresponding homogeneous equation y'' + 3y' + 2y = 0. Find the characteristic equation: r^2 + 3r + 2 = 0. Factorize it to find the roots r = -1 and r = -2. The general solution of the homogeneous equation is y_h = C_1 e^{-x} + C_2 e^{-2x}.

Step 3: Use the method of variation of parameters to find a particular solution y_p for the non-homogeneous equation. The method involves finding two functions u_1(x) and u_2(x) such that y_p = u_1(x)y_1 + u_2(x)y_2, where y_1 = e^{-x} and y_2 = e^{-2x} are solutions of the homogeneous equation.

Step 4: Compute u_1(x) and u_2(x) using the formulas derived from variation of parameters. These formulas involve integrating expressions that include the Wronskian of y_1 and y_2, as well as the non-homogeneous term (1/3 + e^x). After integration, determine the particular solution y_p.

Step 5: Combine the homogeneous solution y_h and the particular solution y_p to form the general solution: y = y_h + y_p = C_1 e^{-x} + C_2 e^{-2x} + 1/6 + (1/2)x e^x.