The differential equation y " + ∂ ( x y ' + y ) = 0 arises in the study of the turbulent flow of a uniform stream past a circular cylinder. Verify that y 1 = exp ( − ∂ x 2 / 2 ) is one solution and then find the general solution in the form of integral.
The differential equation y " + ∂ ( x y ' + y ) = 0 arises in the study of the turbulent flow of a uniform stream past a circular cylinder. Verify that y 1 = exp ( − ∂ x 2 / 2 ) is one solution and then find the general solution in the form of integral.
The differential equation
y
"
+
∂
(
x
y
'
+
y
)
=
0
arises in the study of the turbulent flow of a uniform stream past a circular cylinder. Verify that
y
1
=
exp
(
−
∂
x
2
/
2
)
is one solution and then find the general solution in the form of integral.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Consider the differential equation: dy/dx = y(3 - y)
1) Use the slope field plotter to plot different solutions corresponding to A: (0,-2); B: (0,1); C: (0,3); D: (0,4). You can
drag each point to the correct place if they're somewhere else on the plane, or click on the point and redefine. Capture
a screenshot showing all the solution curves and insert below.
f'(x)=y(3-y)
Kin/5
-5
Density
Length
dy
de
-8
Step size 0.1
Ymas
y (3-9)
Input.
✔Solution A
Solution B
Solution C
Solution D
DCQ=
AC:
2) To what value does y(x) approach, as x → ∞, for the solutions corresponding to B,C, and D? To what value does
y(x) approach, as x →∞, for the solution corresponding to A? Why is it different from the others? Be specific and
answer in terms of what you know about the slope field and its effect on the solution.
-
3) Use wolframalpha.com or something similar to find the particular solution corresponding to each of the points
A,B,C,D. How is the solution through Point C different from the other…
Chapter 4 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
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