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 …

Explain why it is always possible to express any homogeneous D.E. Read more »

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 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  Read more »

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 …

Suppose water is leaking from a tank through a circular hole of area  at its bottom. Read more »

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 …

Find a function whose second derivative is y” = 12x – 2 at each point (x,y) on its graph and y = -x+5 Read more »

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 …

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. Read more »

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 …

Discuss the methods of undetermined coefficients and variation of parameters can be combined to solve the given DE Read more »

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 …

Is the set of functions   linearly dependent or linearly independent on (−∞, ∞)? Read more »

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 the treatment of cancer of the thyroid, the radioactive liquid Iodine-131 is often used 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 …

Suppose a small single-stage rocket of total mass m(t) is launched vertically, the positive direction is upward Read more »