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21. Let y = y₁ (1) be a solution of y' + p(t) y = 0, and let y = y₂(t) be a solution of (22 (28 y' + p(t) y = g(t). Show that y = y₁ (t) + y₂(t) is also a solution of equation (28). 22. a. Show that the solution (7) of the general linear equation (1

21. Let y = y₁ (1) be a solution of y' + p(t) y = 0, and let y = y₂(t) be a solution of (22 (28 y' + p(t) y = g(t). Show that y = y₁ (t) + y₂(t) is also a solution of equation (28). 22. a. Show that the solution (7) of the general linear equation (1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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b. Show that o(t) = 1/t is a solution of y' + y² = 0 fort > 0, but that y = co(t) is not a solution of this equation unless c = 0 or c = 1. Note that the equation of part b is nonlinear, while that of part a is linear. constant c. 20. Show that if y = o(t) is a solution of y' + p(t) y = 0, then y = co(t) is also a solution for any value of the constant c. 21. Let y = y₁ (t) be a solution of y + p(t) y = 0, and let y = y₂(t) be a solution of y = cy₁(t) + y₂(t), y' + p(t) y = g(t). Show that y = y₁ (t) + y2(t) is also a solution of equation (28). 22. a. Show that the solution (7) of the general linear equation (1) can be written in the form (29) where c is an arbitrary constant. b. Show that y₁ is a solution of the differential equation y' + p(t) y = 0, (27) (28) y' + p(t) y = q (t) y", (30) corresponding to g(t) = 0. c. Show that y2 is a solution of the full linear equation (1). We see later (for example, in Section 3.5) that solutions of higher- order linear equations have a pattern similar to equation (29). Bernoulli Equations. Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts form it into a linear equation. The most important such equation has the 25. y' = ey study of the stabil Discontinuous C occur in which c discontinuities. If to solve the equa the two solutions is accomplished following two pro it is impossible al 26. Solve the in where 27. Solve the in where 2.5 Autonomous Diffe and Population Dynam does not appear explicitly. Such equ An important class of first-order equations
Transcribed Image Text:b. Show that o(t) = 1/t is a solution of y' + y² = 0 fort > 0, but that y = co(t) is not a solution of this equation unless c = 0 or c = 1. Note that the equation of part b is nonlinear, while that of part a is linear. constant c. 20. Show that if y = o(t) is a solution of y' + p(t) y = 0, then y = co(t) is also a solution for any value of the constant c. 21. Let y = y₁ (t) be a solution of y + p(t) y = 0, and let y = y₂(t) be a solution of y = cy₁(t) + y₂(t), y' + p(t) y = g(t). Show that y = y₁ (t) + y2(t) is also a solution of equation (28). 22. a. Show that the solution (7) of the general linear equation (1) can be written in the form (29) where c is an arbitrary constant. b. Show that y₁ is a solution of the differential equation y' + p(t) y = 0, (27) (28) y' + p(t) y = q (t) y", (30) corresponding to g(t) = 0. c. Show that y2 is a solution of the full linear equation (1). We see later (for example, in Section 3.5) that solutions of higher- order linear equations have a pattern similar to equation (29). Bernoulli Equations. Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts form it into a linear equation. The most important such equation has the 25. y' = ey study of the stabil Discontinuous C occur in which c discontinuities. If to solve the equa the two solutions is accomplished following two pro it is impossible al 26. Solve the in where 27. Solve the in where 2.5 Autonomous Diffe and Population Dynam does not appear explicitly. Such equ An important class of first-order equations
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