Problem 4. (8 points) In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y"+2y-3y=2e31 – 24te¯31. +31-2 with initial values y(0) = 0 and y(0) = -12. A. Write the characteristic equation for the associated homoge- neous equation. (Use r for your variable.) B. Write the fundamental solutions for the associated homoge- neous equation. y₁ =— y2 C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undeter- mined coefficients. Y' Y Y" Problem 5. (3 points) Solve the following differential equation by variation of ters. Fully evaluate all integrals. y"+9y=sec(3x). parame- (1) Find the most general solution to the associated homoge- neous differential equation. Use c₁ and c₂ in your answer to denote arbitrary constants, and enter them as c1 and c2. help (formulas) Yh (2) Find a particular solution to the nonhomogeneous differ- ential equation y" +9y= sec(3x). Ур help (formulas) (3) Find the most general solution to the original nonho- mogeneous differential equation. Use c₁ and c₂ in your answer to denote arbitrary constants. D. Write the general solution. (Use cl and c2 for c₁ and c₂). y= y= E. C2 Plug in the initial values and solve for c₁ and to find the solution to the initial value problem. Problem 6. (2 points) The IVP y d²x sin(t) dr help (formulas) dx +cos(t). +sin(t)x = tan(t) Hint: No fractions are required in the solution or answer to this problem. dt x(0.5)=19 dx dt 0.5 = 8 has a unique solution defined on the interval. Problem 7. (5 points) Use the Wronskian to determine whether the functions y₁ = sin(2x) and y2 = cos(5x) are linearly independent.
Problem 4. (8 points) In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y"+2y-3y=2e31 – 24te¯31. +31-2 with initial values y(0) = 0 and y(0) = -12. A. Write the characteristic equation for the associated homoge- neous equation. (Use r for your variable.) B. Write the fundamental solutions for the associated homoge- neous equation. y₁ =— y2 C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undeter- mined coefficients. Y' Y Y" Problem 5. (3 points) Solve the following differential equation by variation of ters. Fully evaluate all integrals. y"+9y=sec(3x). parame- (1) Find the most general solution to the associated homoge- neous differential equation. Use c₁ and c₂ in your answer to denote arbitrary constants, and enter them as c1 and c2. help (formulas) Yh (2) Find a particular solution to the nonhomogeneous differ- ential equation y" +9y= sec(3x). Ур help (formulas) (3) Find the most general solution to the original nonho- mogeneous differential equation. Use c₁ and c₂ in your answer to denote arbitrary constants. D. Write the general solution. (Use cl and c2 for c₁ and c₂). y= y= E. C2 Plug in the initial values and solve for c₁ and to find the solution to the initial value problem. Problem 6. (2 points) The IVP y d²x sin(t) dr help (formulas) dx +cos(t). +sin(t)x = tan(t) Hint: No fractions are required in the solution or answer to this problem. dt x(0.5)=19 dx dt 0.5 = 8 has a unique solution defined on the interval. Problem 7. (5 points) Use the Wronskian to determine whether the functions y₁ = sin(2x) and y2 = cos(5x) are linearly independent.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
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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Question
Transcribed Image Text:Problem 4. (8 points)
In this problem you will use undetermined coefficients to solve
the nonhomogeneous equation
y"+2y-3y=2e31 – 24te¯31.
+31-2
with initial values y(0) = 0 and y(0) = -12.
A. Write the characteristic equation for the associated homoge-
neous equation. (Use r for your variable.)
B. Write the fundamental solutions for the associated homoge-
neous equation. y₁ =— y2
C. Write the form of the particular solution and
its derivatives. (Use A, B, C, etc. for undeter-
mined coefficients.
Y'
Y
Y"
Problem 5. (3 points)
Solve the following differential equation by variation of
ters. Fully evaluate all integrals.
y"+9y=sec(3x).
parame-
(1) Find the most general solution to the associated homoge-
neous differential equation. Use c₁ and c₂ in your answer
to denote arbitrary constants, and enter them as c1 and c2.
help (formulas)
Yh
(2) Find a particular solution to the nonhomogeneous differ-
ential equation y" +9y= sec(3x).
Ур
help (formulas)
(3) Find the most general solution to the original nonho-
mogeneous differential equation. Use c₁ and c₂ in your
answer to denote arbitrary constants.
D. Write the general solution. (Use cl and c2 for c₁ and c₂).
y=
y=
E.
C2
Plug in the initial values and solve for c₁ and
to find the solution to the initial value problem.
Problem 6. (2 points)
The IVP
y
d²x
sin(t) dr
help (formulas)
dx
+cos(t). +sin(t)x = tan(t)
Hint: No fractions are required in the solution or answer to this
problem.
dt
x(0.5)=19
dx
dt 0.5
= 8
has a unique solution defined on the interval.
Problem 7. (5 points)
Use the Wronskian to determine whether the functions y₁ =
sin(2x) and y2 = cos(5x) are linearly independent.
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