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

Previous problemNext problem

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Solve the differential equation $y^{''} + 2 y^{'} = 2 x + 9 - e^{- 2 x}$ using the method of undetermined coefficients. Which of the following is a particular solution?

y_{p} = 2 x + 9 + e^{- 2 x}

y_{p} = x^{2} + 9 x - e^{- 2 x}

y_{p} = x + \frac{9}{2} - \frac{1}{2} e^{- 2 x}

y_{p} = a x^{2} + b x + c + d e^{- 2 x}

Previous problemNext problem

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: y'' + 2y' = 2x + 9 - e^{-2x}. The right-hand side is a combination of polynomial, constant, and exponential terms, which suggests using the method of undetermined coefficients.

Step 2: Analyze the form of the particular solution. The method of undetermined coefficients involves guessing a form for the particular solution based on the right-hand side of the equation. For the polynomial term 2x + 9, we guess a solution of the form ax + b. For the exponential term -e^{-2x}, we guess a solution of the form ce^{-2x}. Combine these guesses into a single form: y_p = ax + b + ce^{-2x}.

Step 3: Substitute the guessed particular solution into the differential equation. Compute the first derivative y_p' and second derivative y_p'' of the guessed solution y_p = ax + b + ce^{-2x}. Substitute y_p, y_p', and y_p'' into the original equation y'' + 2y' = 2x + 9 - e^{-2x}.

Step 4: Solve for the coefficients a, b, and c. After substitution, equate the coefficients of like terms (polynomial, constant, and exponential) on both sides of the equation. This will yield a system of equations for a, b, and c. Solve this system to determine the values of a, b, and c.

Step 5: Write the particular solution. Once the coefficients a, b, and c are determined, substitute them back into the guessed form y_p = ax + b + ce^{-2x} to obtain the particular solution. Verify that this solution satisfies the original differential equation.