Consider the initial value problem (a) Find the solution u(t) of this problem. u(t) = b)...
1 (a) For arbitrary real s find the exact solution of the initial value problem with y(0)s>0. (b) Show that the solution blows up when t log(1 +1/s2).
Let u be the solution to the initial boundary value problem for the Heat Equation, фа(t, x)-5 &n(t, x), t E (0,00), x E (0, 1); with initial condition 2 r-, 1 and with boundary condition:s n(t, 0)=0, rn(t, 1-0. Find the solution u using the expansion with the normalization conditions vn (0)-1, wn a. (3/10) Find the functions wz, with indexn> 1 b. (3/10) Find the functions v, with index n> 1. c. (4/10) Find the coefficients cn, with...
4. [10] Find the solution to given initial-boundary value problem: 4uxx = ut 0<x<TI, t> 0 u(0,t) = 5, uit, t) = 10, t> 0 u(x,0) = sin 3x - sin 5x, 0<x<T
Consider the initial value problem dy 3 2- y = 3t + 2e', y(0) = yo . and for yo > Ye, (a) Find the critical value of yo, yc, such that for yo < yc, limt 400 y(t) = - limt700 y(t) = 0. (b) What happens if yo = ye?
-). 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).
5. Given y,-t is a solution of t2y" + 2t1-2y = 0, for t > 0, find a second solution y2.
7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x < 1, t > 0; u(0,t) 0, u1,t 0, t>0; In Problem 10, consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t0. Find an expression for the temperature u(,t) if the initial temperature distribution in the rod is the given function. Suppose that a
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. I. Determine conditional density of the residual lifetime, T-u, given that T 〉 u. 2. Find conditional expectation, E TT>u
4. (10 points) Find the solution to the wave problem Ut = c+421 +COSI, <0, t>0, with initial conditions u(1,0) = sin r, 4(1,0) = 1+I.
the value of resistor is 1k Consider the circuit shown in the f is no initial energy stored in the circuit, and the switch Si closed at time to. A. Find the time constant, t. B. What is the final (steady-state ) current in the circuit C. Find the time it takes for the current (lu) to reach 90% of its final value If the initial current in the inductor equal lo=-15 mA, D. Find the exponintial function that descripe...