Solve the differential equation by separation of variables: . Which of the following is the general solution?
Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
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- 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
- 10. Physics Applications of Integrals 3h 16m
- 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
- 16. Parametric Equations & Polar Coordinates7h 58m
13. Intro to Differential Equations
Basics of Differential Equations
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Join thousands of students who trust us to help them ace their exams!Watch the first videoMultiple Choice
Solve the differential equation using the method of variation of parameters. Which of the following is the general solution?
A
B
C
D

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Step 1: Identify the type of differential equation. The given equation y'' + 3y' + 2y = cos(e x) is a second-order linear non-homogeneous differential equation. The method of variation of parameters is suitable for solving this type of equation.
Step 2: Solve the corresponding homogeneous equation y'' + 3y' + 2y = 0. Find the characteristic equation, which is r^2 + 3r + 2 = 0. Factorize the characteristic equation to find the roots r = -1 and r = -2. The general solution to the homogeneous equation is y_h(x) = C_1 e^{-x} + C_2 e^{-2x}, where C_1 and C_2 are constants.
Step 3: Use the method of variation of parameters to find a particular solution y_p(x) for the non-homogeneous equation. The method involves finding two functions u_1(x) and u_2(x) such that y_p(x) = u_1(x) e^{-x} + u_2(x) e^{-2x}. These functions are determined by solving a system of equations derived from the original differential equation.
Step 4: Compute u_1(x) and u_2(x). First, set up the system of equations: u_1'(x)e^{-x} + u_2'(x)e^{-2x} = 0 and u_1'(x)(-e^{-x}) + u_2'(x)(-2e^{-2x}) = cos(e x). Solve this system for u_1'(x) and u_2'(x), then integrate to find u_1(x) and u_2(x).
Step 5: Combine the homogeneous solution y_h(x) and the particular solution y_p(x) to form the general solution y(x). Substitute the expressions for u_1(x) and u_2(x) into y_p(x), and add y_h(x) to obtain the final general solution: y(x) = C_1 e^{-x} + C_2 e^{-2x} + (specific terms involving cos(e x) and sin(e x)).
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