Question: In Problems 3–8, determine whether the given function is a solution to the given differential equation.
The given function is a solution to the given differential equation.
Firstly, we will differentiate with respect to x,
Again, differentiating the given function with respect to x,
Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,
which is the same as the R.HS. (Right-hand side) of the given differential equation.
Hence, is a solution to the differential equation .
Pendulum with Varying Length. A pendulum is formed by a mass m attached to the end of a wire that is attached to the ceiling. Assume that the length l(t)of the wire varies with time in some predetermined fashion. If
U(t) is the angle in radians between the pendulum and the vertical, then the motion of the pendulum is governed for small angles by the initial value problem where g is the acceleration due to gravity. Assume that where is much smaller than . (This might be a model for a person on a swing, where the pumping action changes the distance from the center of mass of the swing to the point where the swing is attached.) To simplify the computations, take g = 1. Using the Runge– Kutta algorithm with h = 0.1, study the motion of the pendulum when . In particular, does the pendulum ever attain an angle greater in absolute value than the initial angle ?
Stefan’s law of radiation states that the rate of change in the temperature of a body at T (t) kelvins in a medium at M (t) kelvins is proportional to . That is, where K is a constant. Let and assume that the medium temperature is constant, M (t) = 293 kelvins. If T (0) = 360 kelvins, use Euler’s method with h = 3.0 min to approximate the temperature of the body after
(a) 30 minutes.
(b) 60 minutes.
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