Apply Euler's method twice to approximate the solution to the initial value problem on the interval | 0, 1 size h = 0.1. Compare the three-decimal-place values of the two approximations at x = =wit 2 y' =y, y(0) = 3, y(x) = 3ex first with step size h = 0.25, then with step (1) with the value of y of the actual solution.

A thin bar of length L= 3 meters is situated along the x axis so that one end is at x = 0 and the other end is at x = 3. The thermal diffusivity of the bar is k = 0.4. The bar's initial temperature is given by the function f(x) = 45x degrees Celsius. The ends of the bar (x = 0 and x = 3) are then put in an icy bath and kept at a constant 0 degrees C. Let u(x, t) be the temperature in the bar at x at time t, with t measured in seconds. Find u(x, t) and then u; (2, 0.1). Put uz (2, 0.1) calculated accurately to the nearest thousandth (3 decimal places) in the answer box.

t3y'''+3t2y''=1.  Use Variation of Parameters to solve this problem. Three solutions to the homogenous equatoin associated with this is (1/t), 1, and t

A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the other end is at x = 3. The thermal diffusivity of the bar is k = 0.4.  The bar’s initial temperature is given by the function f(x) = 200 - 30x degrees Celsius.  The ends of the bar (x = 0 and x = 3) are then put in an icy bath and kept at a constant 0 degrees C.  Let u(x,t)   be the temperature in the bar at x at time t, with t measured in seconds. Find u(x,t) and then u5(2, 0.1).