<---- equation (6)
<---- equation (6) Using the equation (6) in page 569 of our text, obtain the solution of ut-CUx -lx in integral form satisfying the initial condition u(x, 0) = e- 30 1 | [A(p) cos px + B(p)sin px...
Part A is already done. _ [2 sin(t) o (0)-5.5 a. Form the complementary solution to the homogeneous equation e(-t) 5e (t) + c2 en(-t) en(t) b. Construct a particular solution by assuming the form p(t) (sin t)ã + (cos t)band solving for the undetermined ja + (cost)¡ and solving for constant vectors ã and B. Ep(t)- c. Form the general solution ¢(t) (t) + zp(t) and impose the initial condition to obtain the solution of the initial value problem...
1. Find the particular solution of the differential equation dydx+ycos(x)=2cos(x)dydx+ycos(x)=2cos(x) satisfying the initial condition y(0)=4y(0)=4. 2. Solve the following initial value problem: 8dydt+y=32t8dydt+y=32t with y(0)=6.y(0)=6. (1 point) Find the particular solution of the differential equation dy + y cos(x) = 2 cos(z) satisfying the initial condition y(0) = 4. Answer: y= 2+2e^(-sin(x)) Your answer should be a function of x. (1 point) Solve the following initial value problem: dy ty 8 at +y= 32t with y(0) = 6. (Find y as...
6. Solve the heat equation (5.17) with initial condition u(x, 0) = H(x)e-x. Write the solution of the Cauchy problem for the heat equation u = kuyx - < x <®, t> 0, (5.17) with initial condition u(t,0) = {(H(x + 1) - H (1 - x)) in terms of the error function Erf () = * e ** dy.
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t) Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
MATH2018 Quiz The PDE ar2 can be solved using D'Alembert method. That is, it has a solution of the form u(x, t) = φ(x + ct) + ψ(x-ct). where c 6 Solve the PDE with the initial conditions u(x, 0) 6 sin (x), ut (x, 0) 3 er Enter the expression for u(x, t) in the box below using Maple syntax. Note: the expression should be in terms of x andt, but not c MATH2018 Quiz The PDE ar2 can...
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) =...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1. Let...
Let u be the solution to the initial boundary value problem for the Heat Equation, Otu(t, x) = 2 &n(t, x), ț e (0,00), x e (0,5); with initial condition u(0,xf(x)- and with boundary condition:s Find the solution u using the expansion with the normalization conditions (2n - 1) a. (3/10) Find the functions w, with indexn>1. Wnsin(2n-1)pix/10) b. (3/10) Find the functions v, with indexn > 1. Vnexp(-2(2n-1)pi/10)(2)t) 1. C. (4/10) Find the coefficients cn , with index n...
By 5. (a) Verify that y = {24 sin x is a solution to the differential equation dx2 dy + 5y = 0. dc [10 marks) (b) Differentiate the following functions with respect to c: (i) In(1 + sin? 2) (ii) * 2x3 - 4 - 8 dc. (c) Evaluate the integral / 272 * +432 – 4.7" [15 marks] [25 marks] 6. (a) let f: R+R be a function defined by f(x) 3 + 4 if : 51 ax+b...