In general for a non-homogeneous problem y" + p(z)y +q(z)y = f(z) assume that y1, 2 is a fundamental set of solutions for the homogeneous problem y" + p(z)y' + g(z)y = 0. Then the formula for the particular solution using the method of variation of parameters is Yp = 91u1 + Yu -y2(z)f(z) W(z) 1 (z)f(z) W(z) where u = and u = where W(z) is the Wronskian given by the determinant |31 (z) 2(z) | W(z) = -2(z)f(z) W(z) Y1 (z)f(z) dz. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. W(e) So we have u1 = dz and ug = In other words we have the single integral formula -2(z)f(z) p(z) = y1(z) | W{x) dz + y2(z) dr W(z) As a specific example we consider the non-homogeneous problem y" + 15y + 50y = 25 sin(ez) (1) The general solution of the homogeneous problem (called the complementary solution, ye = ayı + byz ) is given in terms of a pair of linearly independent solutions, y1. Y2. Here a and b are arbitrary constants. Find a fundamental set for y" + 15y + 50y = 0 and enter your results as a comma separated list BEWARE Notice that the above set does not require you to decide which function is to be called y1 or y2 and normally the order you name them is irrelevant. But for the method of variation of parameters an order must be chosen and you need to stick to that order. In order to more easily allow WeBWorK to grade your work I have selected a particular order for y1 and y2. In order to ascertain the order you need to use please enter a choice for y1 = and if your answer is marked as incorrect simply enter the other function from the complementary set. Once you get this box marked as correct then y2 = With this appropriate order we are now ready to apply the method of variation of parameters. (2) For our particular problem we have W()
In general for a non-homogeneous problem y" + p(z)y +q(z)y = f(z) assume that y1, 2 is a fundamental set of solutions for the homogeneous problem y" + p(z)y' + g(z)y = 0. Then the formula for the particular solution using the method of variation of parameters is Yp = 91u1 + Yu -y2(z)f(z) W(z) 1 (z)f(z) W(z) where u = and u = where W(z) is the Wronskian given by the determinant |31 (z) 2(z) | W(z) = -2(z)f(z) W(z) Y1 (z)f(z) dz. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. W(e) So we have u1 = dz and ug = In other words we have the single integral formula -2(z)f(z) p(z) = y1(z) | W{x) dz + y2(z) dr W(z) As a specific example we consider the non-homogeneous problem y" + 15y + 50y = 25 sin(ez) (1) The general solution of the homogeneous problem (called the complementary solution, ye = ayı + byz ) is given in terms of a pair of linearly independent solutions, y1. Y2. Here a and b are arbitrary constants. Find a fundamental set for y" + 15y + 50y = 0 and enter your results as a comma separated list BEWARE Notice that the above set does not require you to decide which function is to be called y1 or y2 and normally the order you name them is irrelevant. But for the method of variation of parameters an order must be chosen and you need to stick to that order. In order to more easily allow WeBWorK to grade your work I have selected a particular order for y1 and y2. In order to ascertain the order you need to use please enter a choice for y1 = and if your answer is marked as incorrect simply enter the other function from the complementary set. Once you get this box marked as correct then y2 = With this appropriate order we are now ready to apply the method of variation of parameters. (2) For our particular problem we have W()
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 17EQ
Related questions
Question
![In general for a non-homogeneous problem z" + p(z)y + g(z)y= f(x) assume that y1, 2 is a fundamental set of solutions for the homogeneous problem y" + p(z)y' + g(z)y = 0. Then the formula for the particular solution
using the method of variation of parameters is
Yp = Y1u1 + yuz
-y2(z)f(x)
W(z)
Y1 (1) f(x)
W(z)
where u =
and u =
where W(z) is the Wronskian given by the determinant
| 31 (z) 2(z)|
|: (z) ½(x)
W(z) =
-Y2(z)f(x)
W(z)
'yh(포)f(x)
W(z)
So we have u1 =
dz and uz =
dr. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero.
In other words we have the single integral formula
Yp(z) = y1(z) /
-y2(z)f(z)
W(z)
y1 (2)f(x)
dz + ya(z) ] W(2)
dr
As a specific example we consider the non-homogeneous problem y" + 15y' + 50y = 25 sin(ez)
(1) The general solution of the homogeneous problem (called the complementary solution, ye = ayı + byz ) is given in terms of a pair of linearly independent solutions, Yı, Y2. Here a and b are arbitrary constants.
Find a fundamental set for y" + 15y' + 50y = 0 and enter your results as a comma separated list
BEWARE Notice that the above set does not require you to decide which function is to be called y1 or y2 and normally the order you name them is irrelevant. But for the method of variation of parameters an order must be chosen and you
need to stick to that order. In order to more easily allow WeBWork to grade your work I have selected a particular order for y1 and y2. In order to ascertain the order you need to use please enter a choice for yı =
and
if your answer is marked as incorrect simply enter the other function from the complementary set. Once you get this box marked as correct then y2 =
With this appropriate order we are now ready to apply the method of variation of parameters.
(2) For our particular problem we have W(z) =
-/ 2(2)f(z)
W(z)
u1 =
dz
dr =
* y1 (1)f(x)
dz =
And combining these results we arrive at
Yp =
(3) Finally, using a and b for the arbitrary constants in ye, the general solution can then be written as
y = Yc + Yp =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee8fb823-9e56-4d34-a863-64debae68570%2F7e33fbad-ae3a-423b-b640-82d77542032f%2Fm41493_processed.png&w=3840&q=75)
Transcribed Image Text:In general for a non-homogeneous problem z" + p(z)y + g(z)y= f(x) assume that y1, 2 is a fundamental set of solutions for the homogeneous problem y" + p(z)y' + g(z)y = 0. Then the formula for the particular solution
using the method of variation of parameters is
Yp = Y1u1 + yuz
-y2(z)f(x)
W(z)
Y1 (1) f(x)
W(z)
where u =
and u =
where W(z) is the Wronskian given by the determinant
| 31 (z) 2(z)|
|: (z) ½(x)
W(z) =
-Y2(z)f(x)
W(z)
'yh(포)f(x)
W(z)
So we have u1 =
dz and uz =
dr. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero.
In other words we have the single integral formula
Yp(z) = y1(z) /
-y2(z)f(z)
W(z)
y1 (2)f(x)
dz + ya(z) ] W(2)
dr
As a specific example we consider the non-homogeneous problem y" + 15y' + 50y = 25 sin(ez)
(1) The general solution of the homogeneous problem (called the complementary solution, ye = ayı + byz ) is given in terms of a pair of linearly independent solutions, Yı, Y2. Here a and b are arbitrary constants.
Find a fundamental set for y" + 15y' + 50y = 0 and enter your results as a comma separated list
BEWARE Notice that the above set does not require you to decide which function is to be called y1 or y2 and normally the order you name them is irrelevant. But for the method of variation of parameters an order must be chosen and you
need to stick to that order. In order to more easily allow WeBWork to grade your work I have selected a particular order for y1 and y2. In order to ascertain the order you need to use please enter a choice for yı =
and
if your answer is marked as incorrect simply enter the other function from the complementary set. Once you get this box marked as correct then y2 =
With this appropriate order we are now ready to apply the method of variation of parameters.
(2) For our particular problem we have W(z) =
-/ 2(2)f(z)
W(z)
u1 =
dz
dr =
* y1 (1)f(x)
dz =
And combining these results we arrive at
Yp =
(3) Finally, using a and b for the arbitrary constants in ye, the general solution can then be written as
y = Yc + Yp =
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning