-x?, x< 0 y = x?, x20 is a solution of the differential equation xy' – 2y = 0 on (-x, 0). 30. In Example 7 we saw that y = ¢1(x) = /25 – x² and y = 62(x) interval (-5, 5). Explain why the piecewise-defined function x* are solutions of dy/dx = -x/y on the -V25 al SV25 – x² -V25 – x², -5

Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
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Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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-x?, x< 0
y =
x?, x20
is a solution of the differential equation xy' – 2y = 0 on
(-x, 0).
30. In Example 7 we saw that y = ¢1(x) = /25 – x² and
y = 62(x)
interval (-5, 5). Explain why the piecewise-defined function
x* are solutions of dy/dx = -x/y on the
-V25
al
SV25 – x²
-V25 – x²,
-5 <x< 0
0 <x< 5
is not a solution of the differential equation on the interval (-5, 5).
In Problems 31–34 find values of m so that the function y = emx is a
solution of the given differential equation.
31. y' + 2y
32. 5y' = 2y
33. y" — 5y' + бу — 0
34. 2y" + 7y' – 4y = 0
at
In Problems 35 and 36 find values of m so that the function y = x" is
a solution of the given differential equation.
of
35. ху" + 2у' %3Dо
36. ху" — 7ху' + 15у 3 0
In Problems 37–40 use the concept that y = c, -∞ < x < ∞, is a
constant function if and only if y' = 0 to determine whether the given
differential equation possesses constant solutions.
38. y' = y + 2y – 3
37. Зху' + 5у
10
39. (y – 1)y' = 1
40. y" + 4y' + бу — 10
In Problems 41 and 42 verify that the indicated pair of functions is a
solution of the given system of differential equations on the interval
(-00, 0).
d?x
dx
= x + 3y
42.
4y + e'
41.
dt?
dt
d²y
4х —
dy
5х + Зy;
dt
e';
dt?
- 2t
+ 3e6r,
x = cos 2t + sin 2t + e',
х— е
- 2t
y = -e
+ 5e6t
- sin 2t – e'
y = -cos 2t
Discussion Problems
43. Make up a differential equation that does not possess any real
solutions.
44. Make up a differential equation that you feel confident
nossesses onlu the trivial solution v – 0 Evnloin vour.
Transcribed Image Text:-x?, x< 0 y = x?, x20 is a solution of the differential equation xy' – 2y = 0 on (-x, 0). 30. In Example 7 we saw that y = ¢1(x) = /25 – x² and y = 62(x) interval (-5, 5). Explain why the piecewise-defined function x* are solutions of dy/dx = -x/y on the -V25 al SV25 – x² -V25 – x², -5 <x< 0 0 <x< 5 is not a solution of the differential equation on the interval (-5, 5). In Problems 31–34 find values of m so that the function y = emx is a solution of the given differential equation. 31. y' + 2y 32. 5y' = 2y 33. y" — 5y' + бу — 0 34. 2y" + 7y' – 4y = 0 at In Problems 35 and 36 find values of m so that the function y = x" is a solution of the given differential equation. of 35. ху" + 2у' %3Dо 36. ху" — 7ху' + 15у 3 0 In Problems 37–40 use the concept that y = c, -∞ < x < ∞, is a constant function if and only if y' = 0 to determine whether the given differential equation possesses constant solutions. 38. y' = y + 2y – 3 37. Зху' + 5у 10 39. (y – 1)y' = 1 40. y" + 4y' + бу — 10 In Problems 41 and 42 verify that the indicated pair of functions is a solution of the given system of differential equations on the interval (-00, 0). d?x dx = x + 3y 42. 4y + e' 41. dt? dt d²y 4х — dy 5х + Зy; dt e'; dt? - 2t + 3e6r, x = cos 2t + sin 2t + e', х— е - 2t y = -e + 5e6t - sin 2t – e' y = -cos 2t Discussion Problems 43. Make up a differential equation that does not possess any real solutions. 44. Make up a differential equation that you feel confident nossesses onlu the trivial solution v – 0 Evnloin vour.
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