Differential Equations with Boundary-Value Problems (MindTap Course List)
9th Edition
ISBN: 9781305965799
Author: Dennis G. Zill
Publisher: Cengage Learning
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Chapter 12.1, Problem 1E
In Problems 1–16 use separation of variables to find, if possible, product solutions for the given partial differential equation.
1.
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1. Classify the following differential equations as to ODE/PDE, order, degree, linearity,
coefficients type and homogeneity. State the independent variables and unknown functions.
[you can use a table]
2.
azu(x.y)
azu(x,y)
ду?
4.
= 0,
ax2
5. + 1 = () – x²,
+ y2x
[d²x
Lat2
%3D
dx
6.
dy
sin y,
dx
7.
dt
dy
= 1,
dt
d0y
dx=f(x), where
yz = 0
8. Eo a;(x):
d°y
= y,
dx°
S3xy' + xz"
9.
y - z' + y" = 0'
10. (1 — х)у' — 4ху %3D cos x,
+ 4y = 0,
12. t*y(5) – ty" + 6y = 0,
d?y
11. x
dx2
4
dy
%3D
dx
%3D
d'u
13.
dr2
du
+ u cos(r + 1),
dr
d²y
14.
dx2
J1+ ()*,
%3D
15. uxx
= 0,
Uyy
d?R
16.
dt2
k
R2
17. (sin 0)y" – (cos 0)y' = 2 exp(y),
18. * -
- (1-)*+x = 0.
1
5. Use the properties of differential operators to solve for the product of
(D + x²)(xD + 2)y
1.3. Using the change of variables
x=u+ 2v,
y = 3u+1v
show that the linear 2 x 2 system of differential equations
du
dt
dv
dt
P
5u + 8v
-2-2v
can be rewritten as a linear second-order differential equation.
Chapter 12 Solutions
Differential Equations with Boundary-Value Problems (MindTap Course List)
Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...
Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 116 use separation of variables to...Ch. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 18ECh. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 20ECh. 12.1 - In Problems 1726 classify the given partial...Ch. 12.1 - Prob. 22ECh. 12.1 - Prob. 23ECh. 12.1 - Prob. 24ECh. 12.1 - Prob. 25ECh. 12.1 - Prob. 26ECh. 12.1 - In Problems 27 and 28 show that the given partial...Ch. 12.1 - In Problems 27 and 28 show that the given partial...Ch. 12.1 - Verify that each of the products u = XY in (3),...Ch. 12.1 - Prob. 30ECh. 12.1 - Prob. 31ECh. 12.1 - Prob. 32ECh. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 16 a rod of length L coincides with...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - In Problems 710 a string of length L coincides...Ch. 12.2 - Prob. 10ECh. 12.2 - In Problems 11 and 12 set up the boundary-value...Ch. 12.2 - In Problems 11 and 12 set up the boundary-value...Ch. 12.3 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.3 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.3 - Find the temperature u(x, t) in a rod of length L...Ch. 12.3 - Solve Problem 3 if L = 2 and f(x)={x,0x10,1x2.Ch. 12.3 - Suppose heat is lost from the lateral surface of a...Ch. 12.3 - Solve Problem 5 if the ends x = 0 and x = L are...Ch. 12.3 - A thin wire coinciding with the x-axis on the...Ch. 12.3 - Find the temperature u(x, t) for the...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 16 solve the wave equation (1) subject...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - In Problems 710 a string is tied to the x-axis at...Ch. 12.4 - Prob. 11ECh. 12.4 - A model for the motion of a vibrating string whose...Ch. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - The transverse displacement u(x, t) of a vibrating...Ch. 12.4 - Prob. 19ECh. 12.4 - The vertical displacement u(x, t) of an infinitely...Ch. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 1–10 solve Laplace’s equation (1) for...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - In Problems 1–10 solve Laplace’s equation (1) for...Ch. 12.5 - In Problems 110 solve Laplaces equation (1) for a...Ch. 12.5 - Prob. 10ECh. 12.5 - In Problems 11 and 12 solve Laplaces equation (1)...Ch. 12.5 - In Problems 11 and 12 solve Laplaces equation (1)...Ch. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - In Problems 15 and 16 use the superposition...Ch. 12.5 - In Problems 15 and 16 use the superposition...Ch. 12.5 - Prob. 18ECh. 12.5 - Solve the Neumann problem for a rectangle:...Ch. 12.5 - Prob. 20ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 3ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 6ECh. 12.6 - Prob. 7ECh. 12.6 - Prob. 8ECh. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - In Problems 1-12 proceed as in Example 1 to solve...Ch. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - In Problems 13-16 proceed as in Example 2 to solve...Ch. 12.6 - Prob. 15ECh. 12.6 - In Problems 13-16 proceed as in Example 2 to solve...Ch. 12.6 - Prob. 17ECh. 12.6 - Prob. 18ECh. 12.6 - Prob. 19ECh. 12.6 - Prob. 20ECh. 12.7 - In Example 1 find the temperature u(x, t) when the...Ch. 12.7 - Prob. 2ECh. 12.7 - Find the steady-state temperature for a...Ch. 12.7 - Prob. 4ECh. 12.7 - Prob. 5ECh. 12.7 - Prob. 6ECh. 12.7 - Prob. 7ECh. 12.7 - Prob. 8ECh. 12.7 - Prob. 9ECh. 12.7 - Prob. 10ECh. 12.8 - In Problems 1 and 2 solve the heat equation (1)...Ch. 12.8 - Prob. 2ECh. 12.8 - Prob. 3ECh. 12.8 - In Problems 3 and 4 solve the wave equation (2)...Ch. 12.8 - Prob. 5ECh. 12.8 - Prob. 6ECh. 12 - Use separation of variables to find product...Ch. 12 - Use separation of variables to find product...Ch. 12 - Find a steady-state solution (x) of the...Ch. 12 - Give a physical interpretation for the boundary...Ch. 12 - At t = 0 a string of unit length is stretched on...Ch. 12 - Prob. 6RECh. 12 - Find the steady-state temperature u(x, y) in the...Ch. 12 - Find the steady-state temperature u(x, y) in the...Ch. 12 - Prob. 9RECh. 12 - Find the temperature u(x, t) in the infinite plate...Ch. 12 - Prob. 11RECh. 12 - Solve the boundary-value problem 2ux2+sinx=ut, 0 ...Ch. 12 - Prob. 13RECh. 12 - The concentration c(x, t) of a substance that both...Ch. 12 - Prob. 15RECh. 12 - Solve Laplaces equation for a rectangular plate...Ch. 12 - Prob. 17RECh. 12 - Prob. 18RECh. 12 - Prob. 19RECh. 12 - If the four edges of the rectangular plate in...
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- A. Determine the order, degree, and type of the following differential equations. DIFFERENTIAL EQUATIONS 1. (x² + y²) dx + 2xy dy = 0 | 2. + ) + = x² + y² ORDER DEGREE TYPE a²u a²u ax2 дх ду. 3. 4 3. x( + (O - y = 0 4. y'' – 3y' + 2y = 0 5. x(y")³ + (y')ª – y = 0 Adx B. Prove that each equation is a solution of the given differential equation. 1. y = x² + 4x; xº = x² + y dx d²x + k²x = 0 dt2 -2x + 4e*; y'"' – 3y' + 2y = 0 2. x sink; %3! 3. y = 3e C. Solve for the general and particular solution for each initial value problem. dy 4. dx 2; y(1) = 4 4x2–7x dy 5. ; y(1) = 1 dx 3y +2 NOTHING FOLLOWSarrow_forwardQ6. Determine the order, degree and state the linearity of the following differential equations: a) (7y")" -G) = -4x tanx -"y- b)arrow_forwardull Turkcell 10:59 %53 elearning.gau.edu.tr 1/1 Assignment- 2 (Sec. 1.1, 1.2, 1.3, 2.2) 1. State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear: 1 y(4) – Py" + 6y = 0 d'u 2 du + u = cos(r + u) dr dr2 dy d²y dx² %3D \dx d²R k dt? 5 (sin 0)y" - (cos 0)y' = 2 2. Solve the given differential equation by separation of variables. 1 (eI 1)?e- dx I (e 1)³e dy - 0 x(1 + y²)\/² dx = y(1 + x²)/2 , dy 3. Find an explicit solution of the given initial-value problem. dx 1 = 4(x + 1), x(7/4) = 1 dt 2 dy y² sin x², y( 2) - dxarrow_forward
- 1. )Solve the Linear Equation dý Зу x3 2dx 2.) Is it an Exact differential equation or not? y (y? + In5x)dy + (x³ + ½)dxarrow_forward1.find the differential of the following functionsarrow_forward8.11.1 Two Dimensional Heat Flow (Laplace equation in two dimensions): The partial differential equation a'u a'u ..(1) represents two dimensional heat flow.arrow_forward
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