2. Solve the linear homogeneous IVP U+ rtuz = 0, u.1,0) = sinr, -o0<< 0, t>...
differential equations Problem 2 Solve y"+y= ſt/2, if 0 <t<6, if t > 6 y(0) = 6, 7(0) = 8
-). Solve the initial and boundary value problem: uUx=0, TE (0,), t > 0, U (0,t) = u(,t) = 0, >0, u(,0) - cos', 1€ (0,7).
Problem 6 [30 points Use Fourier transform to solve the heat equation U = Ura -o0<x< t> 0 subject to the initial condition -1, 1 u(x,0) = -1 < x < 0 0 < x <1 x € (-00, -1) U (1,00)
Find the length of spiral curve T() = ----- 0 < > < 2”
Let T: P2 --> R2 be the linear transformation such that T(x+1)=(1,1), T(x2)=(1,0) and T(x-1)=(0, 1). Find T(2+x+x2).
5. (20 pts). Solve the following initial-value problem: Ut + 2uuz - 0<x<, 0 <t<oo 0 1 <1 > 1 u(t,0) = Then draw the solution for different values of time.
(1 point) Solve the heat problem with non-homogeneous boundary conditions v (2,t) = (2,t), 0<=<4, t>0 u(0,t) =0, u(4,t) = 2, t>0, ulz,0) = , 0 <I<4. Recall that we find h(2), set u(2,t) = u(2,t) – h(2), solve a heat problem for v(, t) and write u(2,t) = v(2,t) +h(2) Find h() (2) = The solution (I, t) can be written as uz, t) =h(2) + (,t), where (2,t) = »=Ecseh (a) v2,t) = Finally, find limu,t) = t-o
2. Solve the initial-boundary value problem One = 48m2 for 0 < x < 8, t > 0, u(0, t) = u(8,t) = 0 for t > 0, u(2,0) = 2e-4x for 0 < x < 8. (60 pts.)
Solve the following problem อน a( 1,0) = 0, a(2,0) = f(0), 0 < θ <う
Solve the IVP y' + y = f(t), y(0) = 0, where f is the 27-periodic function given by f(t) -1, 0<t<T, <t<21, f(t) = f(t + 27).