2. Find the ger 3. Find the general solution of the equation t2x" - 3tx' + 3x = 4t7. r(t) be independent solutions of the m (+) and a =
2. Find the ger 3. Find the general solution of the equation t2x" - 3tx' + 3x = 4t7. r(t) be independent solutions of the m (+) and a =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 22E
Related questions
Question
Please do number 3
![124
EXERCISES
1. Use the variation of parameters formula to find a particular solution to the
following equations.
a) x" + x = tant.
b) x" - x = tet.
c) x" - x = 1.
2. Second-Order Linear Equations
2. Find the general solution of the equation x"+x' = a, where a is a const
3. Find the general solution of the equation t²x" - 3tx' + 3x = 4t7.
4. Let x₁ = x1(t) and x2 = x2(t) be independent solutions of the
equation "+p(t)x' + q(t)x = 0 on an interval I and let W(t) E
Wronskian of X1 and x2.
d) t²x"-2x = t³.
e) x" + x = 1
f) x" - 2x + x = 1et.
a) By direct differentiation show that the Wronskian satisfies the
ential equation W'(t) = -p(t)W(t).
=
b) Solve this equation for W(t) and show that either W(t) 0
te I, or W(t) is never zero on I.
5. Let Lx = x" + px' + cx, where r
expression. Suppose x
o(t) is
0, '(0) = 1. Show that a par
xp (t) = f(ts)f(s)ds, wh
c) If t = a and t = b are two adjacent zeros of x₁(t), i.e., x₁(a) =
0, show that x2(t) must have a zero between a and b. Hint: U
(b).
=
constant,
to Lo
ion to Lx =
constant.
be
C](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc10cce41-ef51-4545-a724-9736e236d3b7%2F2559e490-d294-447b-912f-40a44c4f1e30%2Fwv9g4m5a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:124
EXERCISES
1. Use the variation of parameters formula to find a particular solution to the
following equations.
a) x" + x = tant.
b) x" - x = tet.
c) x" - x = 1.
2. Second-Order Linear Equations
2. Find the general solution of the equation x"+x' = a, where a is a const
3. Find the general solution of the equation t²x" - 3tx' + 3x = 4t7.
4. Let x₁ = x1(t) and x2 = x2(t) be independent solutions of the
equation "+p(t)x' + q(t)x = 0 on an interval I and let W(t) E
Wronskian of X1 and x2.
d) t²x"-2x = t³.
e) x" + x = 1
f) x" - 2x + x = 1et.
a) By direct differentiation show that the Wronskian satisfies the
ential equation W'(t) = -p(t)W(t).
=
b) Solve this equation for W(t) and show that either W(t) 0
te I, or W(t) is never zero on I.
5. Let Lx = x" + px' + cx, where r
expression. Suppose x
o(t) is
0, '(0) = 1. Show that a par
xp (t) = f(ts)f(s)ds, wh
c) If t = a and t = b are two adjacent zeros of x₁(t), i.e., x₁(a) =
0, show that x2(t) must have a zero between a and b. Hint: U
(b).
=
constant,
to Lo
ion to Lx =
constant.
be
C
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