stic equation. (See Problem 21.) There is also a third case that occurs coefficient of the exponential for the larger root ution approaches +∞o or -∞o as in Figures 3.1.1 of the negative. less often: the solution approaches a constant when one exponent is zero and the other is In Sections 3.3 and 3.4, respectively, we return to the problem of solving the equation ay"+by+cy = 0 when the roots of the characteristic equation either are complex conjugate of the mathematical structure of the solutions of all second-order linear homogeneou or are real and equal. In the meantime, in Section 3.2, we provide a systematic discussio equations. Problems In each of Problems 1 through 6, find the general solution of the given differential equation. (18 mia) 1. y" + 2y' - 3y = 0 2. y" + 3y' + 2y = 0 mis 3. 6y" - y - y = 0 4. y" + 5y' = 0 5. 4y" - 9y = 0 6. y" - 2y' - 2y = 0 In each of Problems 7 through 12, find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior as t increases. G 7. y"+y' - 2y = 0, y(0) = 1, y'(0) = 1 8. y" +4y' + 3y = 0, y(0) = 2, y'(0) = -1 G9. y" + 3y = 0, y(0) = -2, y'(0) = 3 G 10. 2y"+y' - 4y = 0, y(0) = 0, y'(0) = 1 G 11. y" +8y' - 9y = 0, y(1) = 1, y' (1) = 0 G 12. 4y" - y = 0, y(-2) = 1, y'(-2) = -1 13. Find a differential equation whose general solution is y=c₁e²¹ +c₂e-³1. G14 G 1)+182 D value problem 16. Solve the initial va y' (0) = 2. Then find t → ∞. In each of Problems 17 which all solutions tenc of a, if any, for which t→∞. 17. y" (2a - 1) y 18. y"+(3-a)y 19. Consider the in y" + 5y' where 3 > 0. a. Solve the i b. Determine of the solution c. Determine d. Determin- Consider the 20. are constants. a. Find all equation. b. Let ye C Thus Y is F
stic equation. (See Problem 21.) There is also a third case that occurs coefficient of the exponential for the larger root ution approaches +∞o or -∞o as in Figures 3.1.1 of the negative. less often: the solution approaches a constant when one exponent is zero and the other is In Sections 3.3 and 3.4, respectively, we return to the problem of solving the equation ay"+by+cy = 0 when the roots of the characteristic equation either are complex conjugate of the mathematical structure of the solutions of all second-order linear homogeneou or are real and equal. In the meantime, in Section 3.2, we provide a systematic discussio equations. Problems In each of Problems 1 through 6, find the general solution of the given differential equation. (18 mia) 1. y" + 2y' - 3y = 0 2. y" + 3y' + 2y = 0 mis 3. 6y" - y - y = 0 4. y" + 5y' = 0 5. 4y" - 9y = 0 6. y" - 2y' - 2y = 0 In each of Problems 7 through 12, find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior as t increases. G 7. y"+y' - 2y = 0, y(0) = 1, y'(0) = 1 8. y" +4y' + 3y = 0, y(0) = 2, y'(0) = -1 G9. y" + 3y = 0, y(0) = -2, y'(0) = 3 G 10. 2y"+y' - 4y = 0, y(0) = 0, y'(0) = 1 G 11. y" +8y' - 9y = 0, y(1) = 1, y' (1) = 0 G 12. 4y" - y = 0, y(-2) = 1, y'(-2) = -1 13. Find a differential equation whose general solution is y=c₁e²¹ +c₂e-³1. G14 G 1)+182 D value problem 16. Solve the initial va y' (0) = 2. Then find t → ∞. In each of Problems 17 which all solutions tenc of a, if any, for which t→∞. 17. y" (2a - 1) y 18. y"+(3-a)y 19. Consider the in y" + 5y' where 3 > 0. a. Solve the i b. Determine of the solution c. Determine d. Determin- Consider the 20. are constants. a. Find all equation. b. Let ye C Thus Y is F
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
1,5
assume: y= e^rt
Transcribed Image Text:of the cha
n in Figures 3.1.1
uation approaches +∞o or -∞o as
efficient of the exponential for the larger root
equation. (See Problem 21.) There is also a third case that occurs
less often: the solution approaches a constant when one exponent is zero and the other is
In Sections 3.3 and 3.4, respectively, we return to the problem of solving the equation
ay"+by+cy=0 when the roots of the characteristic equation either are complex conjugate
or are real and equal. In the meantime, in Section 3.2, we provide a systematic discussio
of the mathematical structure of the solutions of all second-order linear homogeneou
equations.
Jool
Problems
In each of Problems 1 through 6, find the general solution of the given
differential equation.
neden
1. y" +2y' - 3y = 0
2. y" + 3y + 2y = 0 i (1+1)
3.
6y" - y' - y = 0
babala
4.
y" + 5y' = 0
5. 4y" - 9y = 0
17. y" - (2a - 1) y
18. y" +(3-a) y'
6. y" - 2y' - 2y = 0
In each of Problems 7 through 12, find the solution of the given 19. Consider the in
initial value problem. Sketch the graph of the solution and describe
y" + 5y'
its behavior as t increases.
G
G
2013
G
+1+
1) 4 (1Ema)\+
1)+18
G 8. y" +4y' + 3y = 0, y(0) = 2, y'(0) = -1
9. y" + 3y' = 0, y(0) = -2, y'(0) = 3
G 10. 2y"+y' - 4y = 0, y(0) = 0, y'(0) = 1
11. y" +8y' - 9y = 0, y(1) = 1, y'(1) = 0
G 12. 4y" - y = 0, y(-2) = 1, y'(-2) = -1
13. Find a differential equation whose general solution is
y=c₁e²¹ + c₂e-³1
7. y"+y' - 2y = 0, y(0) = 1, y'(0) = 1
problem
16. Solve the initial va
y'(0) = 2. Then find
t →∞.
In each of Problems 17
which all solutions tend
of a, if any, for which
t →∞.
where 3 > 0.
a. Solve the in
b. Determine
of the solution
c. Determine
d. Determin
20. Consider the
are constants.
a. Find all
equation.
b. Let Ye C
Thus Y is
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