(a) The general solution using Separation of Variables and to show that if we set A = u 2 then d u d t = 1 2 k ( 1 − u 2 M )

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Answer 1

Question

Chapter 10.3, Problem 12E

To determine

(a)

The general solution using Separation of Variables and to show that if we set $A = u^{2}$ then

$\frac{d u}{d t} = \frac{1}{2} k (1 - \frac{u^{2}}{M})$

To determine

(b)

To show that the general solution to Eq (7) is $A (t) = M {(\frac{C e^{(k / \sqrt{M}) t} - 1}{C e^{(k / \sqrt{M}) t} + 1})}^{2}$

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(12 points) Let E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}. (a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such that (x, y, z) (psin cos 0, psin sin 0, p cos) € E. (b) (8 points) Calculate the integral E xyz dV using spherical coordinates.

(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}. Calculate the integral y, f(x, y, z) dV.

(14 points) Let f: R3 R and T: R3. →R³ be defined by f(x, y, z) = ln(x²+ y²+2²), T(p, 0,4)=(psin cos 0, psin sin, pcos). (a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the gradient Vg directly, i.e. without using the chain rule. (b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4). (c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate Vg(r,0,4).

Answer 2

Question

Chapter 10.3, Problem 12E

To determine

(a)

The general solution using Separation of Variables and to show that if we set $A = u^{2}$ then

$\frac{d u}{d t} = \frac{1}{2} k (1 - \frac{u^{2}}{M})$

To determine

(b)

To show that the general solution to Eq (7) is $A (t) = M {(\frac{C e^{(k / \sqrt{M}) t} - 1}{C e^{(k / \sqrt{M}) t} + 1})}^{2}$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 10.3, Problem 12E

To determine

(a)

The general solution using Separation of Variables and to show that if we set $A = u^{2}$ then

$\frac{d u}{d t} = \frac{1}{2} k (1 - \frac{u^{2}}{M})$

To determine

(b)

To show that the general solution to Eq (7) is $A (t) = M {(\frac{C e^{(k / \sqrt{M}) t} - 1}{C e^{(k / \sqrt{M}) t} + 1})}^{2}$

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 10.3, Problem 12E

To determine

(a)

The general solution using Separation of Variables and to show that if we set $A = u^{2}$ then

$\frac{d u}{d t} = \frac{1}{2} k (1 - \frac{u^{2}}{M})$

To determine

(b)

To show that the general solution to Eq (7) is $A (t) = M {(\frac{C e^{(k / \sqrt{M}) t} - 1}{C e^{(k / \sqrt{M}) t} + 1})}^{2}$

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 5

Chapter 10.3, Problem 12E

To determine

(a)

The general solution using Separation of Variables and to show that if we set $A = u^{2}$ then

$\frac{d u}{d t} = \frac{1}{2} k (1 - \frac{u^{2}}{M})$

To determine

(b)

To show that the general solution to Eq (7) is $A (t) = M {(\frac{C e^{(k / \sqrt{M}) t} - 1}{C e^{(k / \sqrt{M}) t} + 1})}^{2}$

Answer 6

Chapter 10.3, Problem 12E

To determine

(a)

The general solution using Separation of Variables and to show that if we set $A = u^{2}$ then

$\frac{d u}{d t} = \frac{1}{2} k (1 - \frac{u^{2}}{M})$

Answer 7

Chapter 10.3, Problem 12E

To determine

(a)

The general solution using Separation of Variables and to show that if we set $A = u^{2}$ then

$\frac{d u}{d t} = \frac{1}{2} k (1 - \frac{u^{2}}{M})$

Answer 8

To determine

(b)

To show that the general solution to Eq (7) is $A (t) = M {(\frac{C e^{(k / \sqrt{M}) t} - 1}{C e^{(k / \sqrt{M}) t} + 1})}^{2}$

Answer 9

To determine

(b)

To show that the general solution to Eq (7) is $A (t) = M {(\frac{C e^{(k / \sqrt{M}) t} - 1}{C e^{(k / \sqrt{M}) t} + 1})}^{2}$

Answer 10

Expert Solution & Answer

Answer 11

Expert Solution & Answer

Answer 12

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Answer 13

Ch. 10.1 - Prob. 1PQ Ch. 10.1 - Prob. 2PQ Ch. 10.1 - Prob. 3PQ Ch. 10.1 - Prob. 4PQ Ch. 10.1 - Prob. 5PQ Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E

(a) The general solution using Separation of Variables and to show that if we set A = u 2 then d u d t = 1 2 k ( 1 − u 2 M )

Solution Summary: The author explains that by substituting A = u 2 and separate the variables to determine the solution for the given equation.

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(a)

The general solution using Separation of Variables and to show that if we set $A = u^{2}$ then

$\frac{d u}{d t} = \frac{1}{2} k (1 - \frac{u^{2}}{M})$

(b)

To show that the general solution to Eq (7) is $A (t) = M {(\frac{C e^{(k / \sqrt{M}) t} - 1}{C e^{(k / \sqrt{M}) t} + 1})}^{2}$

Want to see the full answer?

Chapter 10 Solutions

(a) The general solution using Separation of Variables and to show that if we set A = u 2 then d u d t = 1 2 k ( 1 − u 2 M )

Solution Summary: The author explains that by substituting A = u 2 and separate the variables to determine the solution for the given equation.

Concept explainers

Videos