24. y'= ry - ky2, r> 0 and k > 0. This equation is important in population dynamics and is discussed in detail in Section 2.5. 25. y' = ey- oy³, e > 0 and o > 0. This equation occurs in the study of the stability of fluid flow.
24. y'= ry - ky2, r> 0 and k > 0. This equation is important in population dynamics and is discussed in detail in Section 2.5. 25. y' = ey- oy³, e > 0 and o > 0. This equation occurs in the study of the stability of fluid flow.
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
25,
Transcribed Image Text:ht,
1,
is
d
and 25 deal with equations of this type.
23. a. Solve Bernoulli's equation when n = 0; when n = 1.
b. Show that if n # 0, 1, then the substitution v = y¹-n reduces
Bernoulli's equation to a linear equation. This method of solution
was formulated by Leibniz in 1696.
Iter Jako
In each of Problems 24 through 25, the given equation is a Bernoulli
Problem 23b.
equation. In each case solve it by using the substitution mentioned in
24. y' = ry - ky2, r> 0 and k > 0. This equation is important in
population dynamics and is discussed in detail in Section 2.5.
akob Bernoulli. Problems 23
25. y' = ey- oy³, € > 0 and o> 0. This equation occurs in the
study of the stability of fluid flow.
where
Discontinuous Coefficients. Linear differential equations sometimes
occur in which one or both of the functions p and g have jump
discontinuities. If to is such a point of discontinuity, then it is necessary
to solve the equation separately for t < to and t > to. Afterward,
the two solutions are matched so that y is continuous at to; this
is accomplished by a proper choice of the arbitrary constants. The
following two problems illustrate this situation. Note in each case that
it is impossible also to make y' continuous at to.
26. Solve the initial value problem
y' + 2y = g(t), y(0) = 0,
g(t)=
= {1
f1, 0≤t≤ 1,
t > 1.
27. Solve the initial value problem
y' + p(t) y = 0, y(0) = 1,
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