In general for a non-homogeneous problem y" + p(z)y + q(z)y = f(z) assume that yı, 42 is a fundamental set of solutions for the homogeneous problem y" +p(z)y' + q(z)y = 0. Then the formula for the particular solution using the method of variation of parameters is -y2(z)f(z) W(z) Y1 (z)f(z) W(z) where u - and u = where W(z) is the Wronskian given by the determinant |31(z) 2(z)| |3/ (z) ½(z)| W(z) = -½(z)f(z) W(z) *Y1 (z)f(z) W(z) So we have u = dz and uz = dz. 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) [-2(z)f(z) W(z) y1 (z) f(z) dz + y2(z) dz W(z) As a specific example we consider the non-homogeneous problem y" – 4y' + 4y %3= (1+z*) (1) The general solution of the homogeneous problem (called the complementary solution, yc = 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" – 4y' + 4y = 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 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) = dz = dz = W(z) * Y1 (z)f(x) dz dz = W(z) And combining these results we arrive at Yp (3) Finally, the general solution is y = ye + Yp where ye = ayı + byz where a and b are arbitrary constants. Use the general solution to find the unique solution of the IVP with initial conditions y(0) = 3 and y'(0) = 5. y =
In general for a non-homogeneous problem y" + p(z)y + q(z)y = f(z) assume that yı, 42 is a fundamental set of solutions for the homogeneous problem y" +p(z)y' + q(z)y = 0. Then the formula for the particular solution using the method of variation of parameters is -y2(z)f(z) W(z) Y1 (z)f(z) W(z) where u - and u = where W(z) is the Wronskian given by the determinant |31(z) 2(z)| |3/ (z) ½(z)| W(z) = -½(z)f(z) W(z) *Y1 (z)f(z) W(z) So we have u = dz and uz = dz. 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) [-2(z)f(z) W(z) y1 (z) f(z) dz + y2(z) dz W(z) As a specific example we consider the non-homogeneous problem y" – 4y' + 4y %3= (1+z*) (1) The general solution of the homogeneous problem (called the complementary solution, yc = 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" – 4y' + 4y = 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 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) = dz = dz = W(z) * Y1 (z)f(x) dz dz = W(z) And combining these results we arrive at Yp (3) Finally, the general solution is y = ye + Yp where ye = ayı + byz where a and b are arbitrary constants. Use the general solution to find the unique solution of the IVP with initial conditions y(0) = 3 and y'(0) = 5. y =
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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Transcribed Image Text:In general for a non-homogeneous problem y" + p(z)y' + q(z)y= f(x) assume that y1, y2 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 = Y141 + Y2U2
-y2(z)f(x)
y1 (z)f(z)
where u
and u, =
where W(z) is the Wronskian given by the determinant
W(z)
W(z)
| yn (포) ya(프) |
W(z) =
2(2)f(z)
W(z)
ryn (포)f(x)
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
–42(z)f(x)
- dz + y2(z) | w(z)
W(z)
y1 (2)f(=) dz
Yp(z) = y1 (z) |
As a specific example we consider the non-homogeneous problem y" – 4y' + 4y =
(1+z²)
(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, 42. Here a and b are arbitrary constants.
Find a fundamental set for y" – 4y' + 4y = 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
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 y =
o be called y1 or yp and normally the order you name them is irrelevant. But for the method of variation of parameters an order must be chosen and you
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)f(z)
W(z)
u1 =
dz =
dz =
Y1 (z)f(x)
dr =
dz =
And combining these results we arrive at
Yp =
(3) Finally, the general solution is y = Yc + Yp where ye = ayı + byz where a and b are arbitrary constants. Use the general solution to find the unique solution of the IVP with initial conditions y(0) = 3 and y'(0) = 5.
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