Solution: In each of 1 through 6 determine whether the method of separation of variables

In each of Problems 1 through 6 determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations.. xuxx + ut = 0

In each of Problems 1 through 6 determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations.tuxx + xut = 0

In each of Problems 1 through 6 determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations.

In each of Problems 1 through 6 determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations.

In each of Problems 1 through 6 determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations.uxx + (x + y)uyy = 0

In each of Problems 1 through 6 determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations.uxx + uyy + xu = 0

Find the solution of the heat conduction problem 100uxx = ut, 0 < x < 1, t > 0; u(0, t) = 0, u(1, t) = 0, t > 0; u(x, 0) = sin 2x sin 5x, 0 x 1.

Find the solution of the heat conduction problem uxx = 4ut, 0 < x < 2, t > 0; u(0, t) = 0, u(2, t) = 0, t > 0; u(x, 0) = 2 sin(x/2) sin x + 4 sin 2x, 0 x 2.

Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0C for all t > 0. In each of Problems 9 through 12 find an expression for the temperature u(x, t) if the initial temperature distribution in the rod is the given function. Suppose that 2 = 1.

Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0C for all t > 0. In each of Problems 9 through 12 find an expression for the temperature u(x, t) if the initial temperature distribution in the rod is the given function. Suppose that 2 = 1.

Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0C for all t > 0. In each of Problems 9 through 12 find an expression for the temperature u(x, t) if the initial temperature distribution in the rod is the given function. Suppose that 2 = 1.

Consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0C for all t > 0. In each of Problems 9 through 12 find an expression for the temperature u(x, t) if the initial temperature distribution in the rod is the given function. Suppose that 2 = 1.u(x, 0) = x, 0 < x < 40

Consider again the rod in Problem 9. For t = 5 and x = 20 determine how many terms are needed to find the solution correct to three decimal places. A reasonable way to do this is to find n so that including one more term does not change the first three decimal places of u(20, 5). Repeat for t = 20 and t = 80. Form a conclusion about the speed of convergence of the series for u(x, t).

For the rod in Problem 9: (a) Plot u versus x for t = 5, 10, 20, 40, 100, and 200. Put all of the graphs on the same set of axes and thereby obtain a picture of the way in which the temperature distribution changes with time. (b) Plot u versus t for x = 5, 10, 15, and 20. (c) Draw a three-dimensional plot of u versus x and t. (d) How long does it take for the entire rod to cool off to a temperature of no more than 1C?

Follow the instructions in Problem 14 for the rod in Problem 10.

Follow the instructions in Problem 14 for the rod in Problem 11.

For the rod in Problem 12: (a) Plot u versus x for t = 5, 10, 20, 40, 100, and 200. (b) For each value of t used in part (a) estimate the value of x for which the temperature is greatest. Plot these values versus t to see how the location of the warmest point in the rod changes with time. (c) Plot u versus t for x = 10, 20, and 30. (d) Draw a three-dimensional plot of u versus x and t. (e) How long does it take for the entire rod to cool off to a temperature of no more than 1C?

Let a metallic rod 20 cm long be heated to a uniform temperature of 100C. Suppose that at t = 0 the ends of the bar are plunged into an ice bath at 0C, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Find an expression for the temperature at any point in the bar at any later time. Determine the temperature at the center of the bar at time t = 30 s if the bar is made of (a) silver, (b) aluminum, or (c) cast iron.

For the rod of Problem 18 find the time that will elapse before the center of the bar cools to a temperature of 5C if the bar is made of (a) silver, (b) aluminum, or (c) cast iron.

In solving differential equations, the computations can almost always be simplified by the use of dimensionless variables. (a) Show that if the dimensionless variable = x/L is introduced, the heat conduction equation becomes 2u 2 = L2 2 u t , 0 << 1, t > 0. (b) Since L2/2 has the units of time, it is convenient to use this quantity to define a dimensionless time variable = (2/L2)t. Then show that the heat conduction equation reduces to

Consider the equation auxx but + cu = 0, (i) where a, b, and c are constants. (a) Let u(x, t) = et w(x, t), where is constant, and find the corresponding partial differential equation for w. (b) If b = 0, show that can be chosen so that the partial differential equation found in part (a) has no term in w. Thus, by a change of dependent variable, it is possible to reduce Eq. (i) to the heat conduction equation

The heat conduction equation in two space dimensions is 2 (uxx + uyy) = ut. Assuming that u(x, y, t) = X(x)Y(y)T(t), find ordinary differential equations that are satisfied by X(x), Y(y), and T(t).

The heat conduction equation in two space dimensions may be expressed in terms of polar coordinates as 2 [urr + (1/r)ur + (1/r2 )u ] = ut. Assuming that u(r, , t) = R(r)()T(t), find ordinary differential equations that are satisfied by R(r), (), and T(t).