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Consider the following Laplace equation in a circle 0, (a,0) = f(o), (1) where fle) is...
Consider the Laplace equation on a circle of radius a around the origin of the xy-plane:...
Consider the Laplace equation on a circle of radius a around the origin of the xy-plane: p?u=0, Osr<a, -Isosa. The boundary condition is u(a,0)= p cos?o, with p a positive constant. Find the solution u(r,o) by separation of variables. Require that the solution is finite at r = 0, and that the solution is continuous with a continuous derivative at 0 = Ín. To check your solution, set r = a and 0 = 0. You should get u(a,0) =...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y)...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0,...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
Consider the Laplace equation for a ball of radius R described in spherical coordinates (r, 0) 2 1 cot 72 0= n where...
Consider the Laplace equation for a ball of radius R described in spherical coordinates (r, 0) 2 1 cot 72 0= n where is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w:= cos e given by 2 F" (r) +2rF (r) - n(n + 1) F, (r) = 0, (1 w2)G (w) - 2wG", (w) +n(n +1)G, (w) 0. (n 0,1,2,....
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a...
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a given periodic function with satisfying f (0) = f (2π) and Moreover, u(r,0 is bounded for r s a Which of the following is the (general) solution Select one: A. where for B. where )cos(n)de and for C. where and 2m for n- 1,2,3, D. where Co E R f(0) cos(n0)de and for
Let a >0 Solve the following Laplace's equation...
Consider the following initial value problem. y′ + 5y = { 0 t ≤ 1 10...
Consider the following initial value problem.
y′ + 5y =
{
0
t ≤ 1
10
1 ≤ t < 6
0
6 ≤ t < ∞
y(0) = 4
(a)
Find the Laplace transform of the right hand side of the above
differential equation.
(b)
Let y(t) denote the solution to the above
differential equation, and let Y((s) denote the
Laplace transform of y(t). Find
Y(s).
(c)
By taking the inverse Laplace transform of your answer to (b),
the...
Verify the following using MATLAB 2) (a) Consider the following function f(t)=e"" sinwt u (t (1)...
Verify the following using MATLAB
2) (a) Consider the following function f(t)=e"" sinwt u (t (1) .... Write the formula for Laplace transform. L[f)]=F(6) F(6))e"d Where f(t is the function in time domain. F(s) is the function in frequency domain Apply Laplace transform to equation 1. Le sin cot u()]F(s) Consider, f() sin wtu(t). From the frequency shifting theorem, L(e"f()F(s+a) (2) Apply Laplace transform to f(t). F,(s)sin ot u (t)e" "dt Define the step function, u(t u(t)= 1 fort >0...
6. (Duz, pp.101–107) Laplace on A Square. Consider the Laplace equation on the square [0, 1]2:...
6. (Duz, pp.101–107) Laplace on A Square. Consider the Laplace equation on the square [0, 1]2: JUxx + Uyy = 0; (x, y) € (0,1) (0,1) | u(0,y) = °(y); u(1, y) = xy(x,0) = xy(x, 1) = 0. Use separation of variables to obtain a series solution.
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where...
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where f(t) = e-u, 12 0. Please write the forms of the natural and forced solution for this differential equation. You DO NOT need to solve. (7 points) b. Again consider the differential equation f(t), where f(t) is an input and y(t) is the output (response) of interest. Please write the differential equation in state-space form. (10 points) c. The classical method for solving differential...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 +...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t)...
Consider the Laplace equation on a circle of radius a around the origin of the xy-plane: p?u=0, Osr<a, -Isosa. The boundary condition is u(a,0)= p cos?o, with p a positive constant. Find the solution u(r,o) by separation of variables. Require that the solution is finite at r = 0, and that the solution is continuous with a continuous derivative at 0 = Ín. To check your solution, set r = a and 0 = 0. You should get u(a,0) =...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
Consider the Laplace equation for a ball of radius R described in spherical coordinates (r, 0) 2 1 cot 72 0= n where is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w:= cos e given by 2 F" (r) +2rF (r) - n(n + 1) F, (r) = 0, (1 w2)G (w) - 2wG", (w) +n(n +1)G, (w) 0. (n 0,1,2,....
Let a >0 Solve the following Laplace's equation in the disk: with the boundary conditions Assume that is a given periodic function with satisfying f (0) = f (2π) and Moreover, u(r,0 is bounded for r s a Which of the following is the (general) solution Select one: A. where for B. where )cos(n)de and for C. where and 2m for n- 1,2,3, D. where Co E R f(0) cos(n0)de and for
Let a >0 Solve the following Laplace's equation...
Consider the following initial value problem.
y′ + 5y =
{
0
t ≤ 1
10
1 ≤ t < 6
0
6 ≤ t < ∞
y(0) = 4
(a)
Find the Laplace transform of the right hand side of the above
differential equation.
(b)
Let y(t) denote the solution to the above
differential equation, and let Y((s) denote the
Laplace transform of y(t). Find
Y(s).
(c)
By taking the inverse Laplace transform of your answer to (b),
the...
Verify the following using MATLAB
2) (a) Consider the following function f(t)=e"" sinwt u (t (1) .... Write the formula for Laplace transform. L[f)]=F(6) F(6))e"d Where f(t is the function in time domain. F(s) is the function in frequency domain Apply Laplace transform to equation 1. Le sin cot u()]F(s) Consider, f() sin wtu(t). From the frequency shifting theorem, L(e"f()F(s+a) (2) Apply Laplace transform to f(t). F,(s)sin ot u (t)e" "dt Define the step function, u(t u(t)= 1 fort >0...
6. (Duz, pp.101–107) Laplace on A Square. Consider the Laplace equation on the square [0, 1]2: JUxx + Uyy = 0; (x, y) € (0,1) (0,1) | u(0,y) = °(y); u(1, y) = xy(x,0) = xy(x, 1) = 0. Use separation of variables to obtain a series solution.
Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where f(t) = e-u, 12 0. Please write the forms of the natural and forced solution for this differential equation. You DO NOT need to solve. (7 points) b. Again consider the differential equation f(t), where f(t) is an input and y(t) is the output (response) of interest. Please write the differential equation in state-space form. (10 points) c. The classical method for solving differential...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t)...
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