+3y=t; y(t) = t/3, y₂(t)=e+t/3 9. 12y"+5ty'+4y=0, t> 0; y(t) = t2, y₂(t) = 1-² Int
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
9
Transcribed Image Text:Problems
In each of Problems 1 through 4, determine the order of the given
differential equation; also state whether the equation is linear or
nonlinear.
1. 12d²y
2.
dy
+1
dt² dt
3.
+ 2y = sin t
(1+ y²)
dy
d²y
dt² dt
+t-
+y=e¹
dªy
d³y d²y
dy
+
+
+ +y=1
dt4 dt³ dt² dt
d²y
4.
dt²
In each of Problems 5 through 10, verify that each given function is a
solution of the differential equation.
5. y" -y=0; y(t) = e, y2(t) = cosht
6. y" +2y' - 3y = 0; y₁(t) = e-³1, y₂(t) = et
7. ty' - y = t²; y = 3t+t²
8. y + 4y+3y= t; y₁(t) = t/3, y2(t) = e+t/3
9. 1²y" +5ty' +4y = 0, t> 0; yi(t) = t2, y₂(t) = t-² Int
10. y' - 2¹y = 1; y=e³²
e³ds + e²
+ sin(t + y) = sint
In each of Problems 11 through 13, determine the values of r for which
the given differential equation has solutions of the form yet.
11. y' +2y=0
12. y"+y'-6y=0
13. y" - 3y" + 2y' = 0
In each of Problems 14 and 15, determine the values of r for which the
given differential equation has solutions of the form y = t' for t > 0.
14. ty" +4ty' + 2y = 0
15. 1²y" - 4ty' + 4y = 0
In each of Problems 16 through 18, determine the order of the given
partial differential equation; also state whether the equation is linear
or nonlinear. Partial derivatives are denoted by subscripts.
16.
Uxx + Uyy + Uzz = 0
17.
Uxxxx +2uxxyy + Uyyyy = 0
18. ut + uux = 1 + Uxx
10.
In each of Problems 19 through 21, verify that each given function is
a solution of the given partial differential equation.
19. Uxx + Uyy = 0; u₁(x, y) = cos x cosh y,
u₂(x, y) = ln(x² + y²)
20. a²uxx = ui;
u₂(x, t) = e-α²x²t
u₁(x, t) = e-a²t sinx,
sin λx, À a real constant
21. a²uxx = Utti
u₂(x, t) = sin(x-at), λ a real constant
u₁(x, t) = sin(x) sin(at),
22. Follow the steps indicated here to derive the equation of motion
of a pendulum, equation (12) in the text. Assume that the rod is rigid
and weightless, that the mass is a point mass, and that there is no
friction or drag anywhere in the system.
to
a. Assume that the mass is in an arbitrary displaced position,
indicated by the angle 0. Draw a free-body diagram showing the
forces acting on the mass.
b. Apply Newton's law of motion in the direction tangential to
the circular arc on which the mass moves. Then the tensile force
in the rod does not enter the equation. Observe that you need to
find the component of the gravitational force in the tangential
direction. Observe also that the linear acceleration, as opposed to
the angular acceleration, is Ld²0/dt², where L is the length of
the rod.
c. Simplify the result from part b to obtain equation (12) in the
text.
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