1a.) Find the solution xa of the Bessel equation t2x'' + tx' + t2x = 0 such that xa(0) = a
1a.) Find the solution xa of the Bessel equation t2x'' + tx' + t2x = 0 such that xa(0) = a
For question 1, answer should be in the form xa(t) = aJ0(t).
1b.) Find the solution xa of the Bessel equation t2x'' + tx' + (t2-1)x = 0 such that x'a(0) = a
1c.) Find x(t) = Σk>=0 aktksuch that x'' = tx + 1 and x(0) = 0, x'(0) = 1
Homework Answers
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1a.) Find the solution xa of the Bessel equation t2x'' + tx' + t2x = 0 such that xa(0) = a
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62 This solution is known as Jo(t). (c) F...
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62 This solution is known as Jo(t). (c) Find a second solution using the method of reduction of order
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62...
9. Show that y xwax2) is a solution of the Airy's differential equation y'taxy-0, x> 0whenever...
9. Show that y xwax2) is a solution of the Airy's differential equation y'taxy-0, x> 0whenever w is a solution of the Bessel's equation tw-0,t>0. Hint: After differentiating, substituting,and simplifying, let tx] (10 points)
1. The general form of a Bessel equation of order v (a constant) is ry" +...
1. The general form of a Bessel equation of order v (a constant) is ry" + ry' +(22 - 12)y=0. (Compare it with the general form of an Euler equation). The solutions of a Bessel equation are called cylindrical function or Bessel function. One example of such a function would be the radial part of the modes of vibration of a circular drum. Consider the following Bessel equation with v = 1 2?y" + ry' +(22y = 3rVīsin c. 1...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equation...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
(1 point) Use the indicated change of variable to find the general solution of the differential equation on (0, oo): General solution for w: w -cjJ +2J General solution for y: y- ci NOTE REGARDING AN...
(1 point) Use the indicated change of variable to find the general solution of the differential equation on (0, oo): General solution for w: w -cjJ +2J General solution for y: y- ci NOTE REGARDING ANSWER ENTRY: To enter a Bessel function of the form Ja(bx), you should type a in the first blank and bz in the second blank. Subscripts should be listed in decreasing order, if applicable.
(1 point) Use the indicated change of variable to find the...
1a). Consider the equation ay" + by' + cy = d where d ∈ R and...
1a). Consider the equation ay" + by' + cy = d where d ∈ R and a, b and c are positive constants. Show that any solution of this equation approaches d/c as x → +∞. That is, given y(x) a solution we have lim y(x) as x → +∞ = d/c . 1b.) What happens if c = 0? 1c.) What about the case where b = c = 0?
(1 point) Find the general solution of the differential equation on (0,0): xy" + xy +...
(1 point) Find the general solution of the differential equation on (0,0): xy" + xy + (576x2 – 484)y=0 General solution: y=c] ( )+czy ( ) NOTE REGARDING ANSWER ENTRY: To enter a Bessel function of the form J. (b), you should type a in the first blank and bx in the second blank. Subscripts should be listed in decreasing order, if applicable.
h Bessel equation of order p is ty" + ty + (t? - p2 y =...
h Bessel equation of order p is ty" + ty + (t? - p2 y = 0. In this problem assume that p= 2. a) Show that y1 = sint/Vt and y2 = cost/vt are linearly independent solutions for 0 <t<o. b) Use the result from part (a), and the preamble in Exercise 3, to find the general solution of ty" + ty' + (t2 - 1/4)y = 3/2 cost. (o if 0 <t < 12, y(t) = { 2...
The Bessel equation of order n: Please use the Forbenius method (below for when is a...
The Bessel equation of order n:
Please use the Forbenius method (below for when is a positive
intiger) to find two solutions as a series in x, when n=1/2.
Lastly find these solutions in CLOSED FORM
expressions. Please show all steps. DO NOT use
Bessel functions
.
0= f(,u — ) + fix + R-1
1a) 1b) 1c) 1d) Find polar coordinates of the point that has rectangular coordinates (-2,2). Write...
1a)
1b)
1c)
1d)
Find polar coordinates of the point that has rectangular coordinates (-2,2). Write your answer using degrees. Polar coordinates: (1) 8 aja x 3 ?. Let O be an angle in quadrant IV such that sin e cola Find the exact values of sec 0 and cot 0. 05 sec 0 = x $ ? cot Find polar coordinates of the point that has rectangular coordinates (3/3,-3). Write your answer using degrees. Polar coordinates: ((,1) 05 x...
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62 This solution is known as Jo(t). (c) Find a second solution using the method of reduction of order
23. Consider the Bessel equation of order zero (a) Show thatr 0 is a double root of the indicial equation. (b) Find one solution of the form 4 22 22.42 22.42.62...
9. Show that y xwax2) is a solution of the Airy's differential equation y'taxy-0, x> 0whenever w is a solution of the Bessel's equation tw-0,t>0. Hint: After differentiating, substituting,and simplifying, let tx] (10 points)
1. The general form of a Bessel equation of order v (a constant) is ry" + ry' +(22 - 12)y=0. (Compare it with the general form of an Euler equation). The solutions of a Bessel equation are called cylindrical function or Bessel function. One example of such a function would be the radial part of the modes of vibration of a circular drum. Consider the following Bessel equation with v = 1 2?y" + ry' +(22y = 3rVīsin c. 1...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
(1 point) Use the indicated change of variable to find the general solution of the differential equation on (0, oo): General solution for w: w -cjJ +2J General solution for y: y- ci NOTE REGARDING ANSWER ENTRY: To enter a Bessel function of the form Ja(bx), you should type a in the first blank and bz in the second blank. Subscripts should be listed in decreasing order, if applicable.
(1 point) Use the indicated change of variable to find the...
(1 point) Find the general solution of the differential equation on (0,0): xy" + xy + (576x2 – 484)y=0 General solution: y=c] ( )+czy ( ) NOTE REGARDING ANSWER ENTRY: To enter a Bessel function of the form J. (b), you should type a in the first blank and bx in the second blank. Subscripts should be listed in decreasing order, if applicable.
h Bessel equation of order p is ty" + ty + (t? - p2 y = 0. In this problem assume that p= 2. a) Show that y1 = sint/Vt and y2 = cost/vt are linearly independent solutions for 0 <t<o. b) Use the result from part (a), and the preamble in Exercise 3, to find the general solution of ty" + ty' + (t2 - 1/4)y = 3/2 cost. (o if 0 <t < 12, y(t) = { 2...
The Bessel equation of order n:
Please use the Forbenius method (below for when is a positive
intiger) to find two solutions as a series in x, when n=1/2.
Lastly find these solutions in CLOSED FORM
expressions. Please show all steps. DO NOT use
Bessel functions
.
0= f(,u — ) + fix + R-1
1a)
1b)
1c)
1d)
Find polar coordinates of the point that has rectangular coordinates (-2,2). Write your answer using degrees. Polar coordinates: (1) 8 aja x 3 ?. Let O be an angle in quadrant IV such that sin e cola Find the exact values of sec 0 and cot 0. 05 sec 0 = x $ ? cot Find polar coordinates of the point that has rectangular coordinates (3/3,-3). Write your answer using degrees. Polar coordinates: ((,1) 05 x...
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