Explain why it is always possible to express any homogeneous D.E. in the form you might start by proving that: Any homogeneous O.D.E can be written as because when identifying our functions of F(x,y) such that: because our M corresponds to y …
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Suppose P(x) is continuous on some interval I and a is a number in I. What can be said about the solution of the initial-value problem ? As always, first, I would like to write down the given data. Sometimes, we have …
Suppose water is leaking from a tank through a circular hole of area at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second …
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Find a function whose second derivative is y” = 12x – 2 at each point (x,y) on its graph and y = -x+5 is tangent to the graph at the point corresponding to x = 1. This problem has multiple …
20. Suppose that y(x) denotes a solution of the first-order IVP and that y(x) possess at least a second derivative at x=1. In some neighborhood of x=1 use the DE to determine whether y(x) is increasing or decreasing and whether the …
Discuss the methods of undetermined coefficients and variation of parameters can be combined to solve the given DE: The auxiliary equation yields . Therefore . Next the presence of in g(x) suggests that the particular solution includes a linear polynomial and solved …
Is the set of functions linearly dependent or linearly independent on (−∞, ∞)? Discuss. Functions will only be Linearly dependent when the meet these requirements: Within the functions intervals of , we will have multiple constants ‘c’. All of our constants …
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In the treatment of cancer of the thyroid, the radioactive liquid Iodine-131 is often used. Suppose that after one day in storage, analysis shows that an initial amount A0 of iodine-131 in a sample has decreased by 8.3%. (a) Find …
In one model of the changing population P(t) of a community, it is assumed that where dB/dt and dD/dt are the birth and death rates, respectively. (a) Solve for P(t) if dB/dt=k1P and dD/dt=k2P. (b) Analyze the cases k1>k2, …
In one model of the changing population P(t) of a community, it is assumed that Read more »
Suppose a small single-stage rocket of total mass m(t) is launched vertically, the positive direction is upward, the air resistance is linear, and the rocket consumes its fuel at a constant rate. A mathematical model for the velocity v(t) of …