Use the Fourier transform to find a solution of the ordinary differential equation u´´-u+2g(x) =0 where g∈L1. (The solution obtained this way is the one that vanishes at ±∞. What is the general solution?)
Homework Answers
Add Answer to:
Use the Fourier transform to find a solution of the ordinary differential equation u´´-u+2g(x) =0 where g∈L1. (The solu...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
Use the Laplace transform to find the solution to the differential equation y'' + y =...
Use the Laplace transform to find the solution to the differential equation y'' + y = U(t − 1), y(0) = 1, y' (0) = 0. Describe the physical system that this differential equation represents. Plot your solution.
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solutio...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
9. (a) Find the inverse Fourier transform of the following function 1 (2 iw)(5 iw) (b)...
9. (a) Find the inverse Fourier transform of the following function 1 (2 iw)(5 iw) (b) The displacement of a particular mechanical system is governed by the following ordinary differential equation dy 10y f(t) 7 dt where y(t) is the displacement and f(t) is the applied load Page 2 of 4 MATH2124 SaMplE EXAM IV i Use the Fourier transform to obtain the impulse response h(t) of the mechanical system (ii) If the applied load is f(t) = H (t1)-...
2.14. For each differential equation given below, find the solution for t 2 0 with the...
2.14. For each differential equation given below, find the solution for t 2 0 with the specified input signal and subject to the specified initial value. Use the general solution technique outlined in Section 2.5.4. of y (t) dt2 dy (t) dy (t) , た0 dy (t) dt22 t 4-t2 + 3 y (t)-x(t) , dP2+3y(t) =x(t), x(t)=u(t), y(0) = 2 dt22+2dy(2+y(t)=x(t) , x(t) = e-2t u (t), x (t) = (t+ 1) u (t) , y (0)--2 dy (t)...
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equati...
Fourier transform:
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
7. Use a suitable Fourier Transform to find the solution of the IVP 2t-r-1 ,2-1 t 〉 0, , u(x, t)...
Please show all steps to solution.
7. Use a suitable Fourier Transform to find the solution of the IVP 2t-r-1 ,2-1 t 〉 0, , u(x, t), uz (x, t) 0asx→00, t〉0, →
7. Use a suitable Fourier Transform to find the solution of the IVP 2t-r-1 ,2-1 t 〉 0, , u(x, t), uz (x, t) 0asx→00, t〉0, →
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms Fu(x, t) _ u(x, t) + g(x) =-a ди (x, t) The domain of the problem is-oo < x <...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms Fu(x, t) _ u(x, t) + g(x) =-a ди (x, t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u(x, 0) 0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following...
Using Fourier transform, prove that a solution of the Laplace equation in the half plane: Urn+...
Using Fourier transform, prove that a solution of the Laplace equation in the half plane: Urn+ Uyy=0,- << ,y>0, with the boundary conditions u(1,0) = f(t), - <I< u(x,y) +0,31 +0,+0, is given by r(2, y) == Love you > 0. Hint: 1. Take Fourier transform on the variable r, 2. Observe U(k, y) +0 as y → 00, 3. Use pt {e-Mliv = Vice in
Use the substitution x = et to transform the given Cauchy-Euler equation to a differential equation...
Use the substitution
x =
et
to transform the given Cauchy-Euler equation to a differential
equation with constant coefficients. (Use yp for
dy
dt
and ypp for
d2y
dt2
.)
x2y'' +
7xy' − 16y = 0
Use the substitution x = ef to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. (Use yp for dy and ypp for dt dt2 x?y" + 7xy' - 16y = 0 x Solve the original equation by solving the...
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms au(x,t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u (x,0)-0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following ODE where G() is the Fourier transform of g(x) and U(w,...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
9. (a) Find the inverse Fourier transform of the following function 1 (2 iw)(5 iw) (b) The displacement of a particular mechanical system is governed by the following ordinary differential equation dy 10y f(t) 7 dt where y(t) is the displacement and f(t) is the applied load Page 2 of 4 MATH2124 SaMplE EXAM IV i Use the Fourier transform to obtain the impulse response h(t) of the mechanical system (ii) If the applied load is f(t) = H (t1)-...
2.14. For each differential equation given below, find the solution for t 2 0 with the specified input signal and subject to the specified initial value. Use the general solution technique outlined in Section 2.5.4. of y (t) dt2 dy (t) dy (t) , た0 dy (t) dt22 t 4-t2 + 3 y (t)-x(t) , dP2+3y(t) =x(t), x(t)=u(t), y(0) = 2 dt22+2dy(2+y(t)=x(t) , x(t) = e-2t u (t), x (t) = (t+ 1) u (t) , y (0)--2 dy (t)...
Fourier transform:
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
3. Consider the equation a(x, 0) = f(x) u(x,t) lim 0 Using a Fourier transform, solve this equation. Evaluate your solution in the case when f(x)-δ(x).
Please show all steps to solution.
7. Use a suitable Fourier Transform to find the solution of the IVP 2t-r-1 ,2-1 t 〉 0, , u(x, t), uz (x, t) 0asx→00, t〉0, →
7. Use a suitable Fourier Transform to find the solution of the IVP 2t-r-1 ,2-1 t 〉 0, , u(x, t), uz (x, t) 0asx→00, t〉0, →
Problem 1: We are interested in solving a modified form of diffusion equation given below using Fourier transforms Fu(x, t) _ u(x, t) + g(x) =-a ди (x, t) The domain of the problem is-oo < x < oo and is 0 < t < oo . At time t = 0, the initial condition is given by u(x, 0) 0 a) Take the Fourier transform on x and show that the above PDE can be transformed into the following...
Using Fourier transform, prove that a solution of the Laplace equation in the half plane: Urn+ Uyy=0,- << ,y>0, with the boundary conditions u(1,0) = f(t), - <I< u(x,y) +0,31 +0,+0, is given by r(2, y) == Love you > 0. Hint: 1. Take Fourier transform on the variable r, 2. Observe U(k, y) +0 as y → 00, 3. Use pt {e-Mliv = Vice in
Use the substitution
x =
et
to transform the given Cauchy-Euler equation to a differential
equation with constant coefficients. (Use yp for
dy
dt
and ypp for
d2y
dt2
.)
x2y'' +
7xy' − 16y = 0
Use the substitution x = ef to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. (Use yp for dy and ypp for dt dt2 x?y" + 7xy' - 16y = 0 x Solve the original equation by solving the...
Most questions answered within 3 hours.
-
History
Lincoln's religious thought was almost exclusively shaped by his
Calvinist upbringing.
True
False
QUESTION 6...
asked 1 minute from now -
In a two-way factorial ANOVA, the final F-ratio for
factor AxB is determined by dividing _____...
asked 48 seconds from now -
If 1.6g of CH4 reacts with oxygen gas to form water and carbon
dioxide what is...
asked 1 minute from now -
Applying the content from Primal Leadership, and the format
Personal Change Plan on p. 110 Boyatzis...
asked 43 seconds ago -
Your black labrador has a genotype of BbEe.
Which of the following is correct? Select all...
asked 4 minutes ago -
Which of the following statements explains why the cost of
advertising might be relevant to a...
asked 14 minutes ago -
NO3- nitrate ion, what is the symmetry operationof rotation of
it?
asked 19 minutes ago -
Block A, with mass mA, is initially at rest on a frictionless
horizontal floor. Block B,...
asked 20 minutes ago -
Space shuttle astronauts each consume an average of 3000
calories per day. One meal normally consists...
asked 22 minutes ago -
A distribution center for a sporting goods retailer places
orders with manufacturers for a variety of...
asked 37 minutes ago -
Suppose you have a bag of Skittles with 27 blue Skittles, 18
green
Skittles, 32 red...
asked 40 minutes ago -
Sketch the circuit diagram for an inverting amplifier, and give
component values that will produce a...
asked 43 minutes ago