Question 1. Consider the second order partial differential equation c2 (1) for an unknown real-valued function u = u(t, x), where t represents time, r represents a point in space, and c>0 is a constant. 1. For any twice differentiable functions F = satisfies (1). = F(x + ct) + G(x – ct) F(x) and G = G(x), show that u(t, a) Partial differential equations such as (1) the unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are given that u(0, x) = g(x) and (0, x) = h(x), for some given functions g and h. are often solved as initial value problems, where the initial description of 2. Assume that u(t, x) = F(x+ ct) + G(x – ct) for some functions F and G, as described in problem 1.1. If u = u(t, x) solves the initial value problem described above, show that g(x) F(x)+ G(x) and h(x) = cF' (x) – cG' (x). 3. By integrating the last equation for h(x), show that for any constant a ER, | h(s) ds = cF(x) – cG(x) – cF(a) + cG(a), and from here solve a linear system to show that 1 F(x) = (9(=) + h(s) ds + F(a) – G(a) , and G(x) =; (9(2) – / h(s) ds – F(a) + G(a) ). - 4. Lastly, given that u(t, r) value problem for 1): F(x + ct) + G(x – ct), arrive at an explicit formula for the solution to the initial = 1 r+ct u(t, æ) = [g(x + ct)+ g(x – ct)] + 2c x-ct Question 2. Consider the function f(x, y, z) = xye -(²+u²+z²). Find all points (æ, y, z) such that Vf(x, y, z) = %3D (0.0 0)
Question 1. Consider the second order partial differential equation c2 (1) for an unknown real-valued function u = u(t, x), where t represents time, r represents a point in space, and c>0 is a constant. 1. For any twice differentiable functions F = satisfies (1). = F(x + ct) + G(x – ct) F(x) and G = G(x), show that u(t, a) Partial differential equations such as (1) the unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are given that u(0, x) = g(x) and (0, x) = h(x), for some given functions g and h. are often solved as initial value problems, where the initial description of 2. Assume that u(t, x) = F(x+ ct) + G(x – ct) for some functions F and G, as described in problem 1.1. If u = u(t, x) solves the initial value problem described above, show that g(x) F(x)+ G(x) and h(x) = cF' (x) – cG' (x). 3. By integrating the last equation for h(x), show that for any constant a ER, | h(s) ds = cF(x) – cG(x) – cF(a) + cG(a), and from here solve a linear system to show that 1 F(x) = (9(=) + h(s) ds + F(a) – G(a) , and G(x) =; (9(2) – / h(s) ds – F(a) + G(a) ). - 4. Lastly, given that u(t, r) value problem for 1): F(x + ct) + G(x – ct), arrive at an explicit formula for the solution to the initial = 1 r+ct u(t, æ) = [g(x + ct)+ g(x – ct)] + 2c x-ct Question 2. Consider the function f(x, y, z) = xye -(²+u²+z²). Find all points (æ, y, z) such that Vf(x, y, z) = %3D (0.0 0)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve 2 part only and take a thumb up plz solve this perfectly.
![Question 1. Consider the second order partial differential equation
(1)
for an unknown real-valued function u = u(t, x), where t represents time, r represents a point in space, and c> 0 is
a constant.
1. For any twice differentiable functions F =
satisfies 1).
F(x) and G = G(x), show that u(t, x) = F(x + ct) + G(x – ct)
Partial differential equations such as 1)
the unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are
given that u(0, x) = g(x) and (0, x) = h(x), for some given functions g and h.
are often solved as initial value problems, where the initial description of
2. Assume that u(t, x) = F(x + ct) + G(x – ct) for some functions F and G, as described in problem 1.1. If
u = u(t, x) solves the initial value problem described above, show that
g(x) = F(x) + G(x)
and
h(x) = cF' (x) – cG' (x).
3. By integrating the last equation for h(x), show that for any constant a E R,
| h(s) ds = cF(x) – cG(x) – cF(a) + cG(a),
and from here solve a linear system to show that
F(2) = (o(2) +[ h(0) ds + F(a) – G(a).
F(x) =
and
G(x) = ; (9(=) – h(s) ds – F(a) + G(a)
4. Lastly, given that u(t, x) = F(x + ct) + G(x – ct), arrive at an explicit formula for the solution to the initial
value problem for 1):
1
1
(r+ct
u(t, æ) = ; [g(x + et) + g(x – ct)] + h(s)ds.
2c
r-ct
xye-(2²+v°+z*). Find all points (x, y, z) such that Vf(x, y, z) =
Question 2. Consider the function f(x, y, z)
(0,0,0).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b6ea445-fcb8-43e5-8ecc-e13cb9c0bb44%2Fd80b7c32-bee8-4342-930e-9424ee106b18%2Fzamj0v7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 1. Consider the second order partial differential equation
(1)
for an unknown real-valued function u = u(t, x), where t represents time, r represents a point in space, and c> 0 is
a constant.
1. For any twice differentiable functions F =
satisfies 1).
F(x) and G = G(x), show that u(t, x) = F(x + ct) + G(x – ct)
Partial differential equations such as 1)
the unknown function and its time derivative are provided at each point in space. Along these lines, suppose we are
given that u(0, x) = g(x) and (0, x) = h(x), for some given functions g and h.
are often solved as initial value problems, where the initial description of
2. Assume that u(t, x) = F(x + ct) + G(x – ct) for some functions F and G, as described in problem 1.1. If
u = u(t, x) solves the initial value problem described above, show that
g(x) = F(x) + G(x)
and
h(x) = cF' (x) – cG' (x).
3. By integrating the last equation for h(x), show that for any constant a E R,
| h(s) ds = cF(x) – cG(x) – cF(a) + cG(a),
and from here solve a linear system to show that
F(2) = (o(2) +[ h(0) ds + F(a) – G(a).
F(x) =
and
G(x) = ; (9(=) – h(s) ds – F(a) + G(a)
4. Lastly, given that u(t, x) = F(x + ct) + G(x – ct), arrive at an explicit formula for the solution to the initial
value problem for 1):
1
1
(r+ct
u(t, æ) = ; [g(x + et) + g(x – ct)] + h(s)ds.
2c
r-ct
xye-(2²+v°+z*). Find all points (x, y, z) such that Vf(x, y, z) =
Question 2. Consider the function f(x, y, z)
(0,0,0).
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