Page 89 - Schaum's Outline of Differential Equations

72 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 7

7.79. An RL circuit has an emf given (in volts) by 4 sin t, a resistance of 100 ohms, an inductance of 4 henries, and no
initial current. Find the current at any time t.

7.80. The steady-state current in a circuit is known to be -fysin t - -^cos t. Rewrite this current in the form A sin (t— (f>).

7.81. Rewrite the steady-state current of Problem 7.21 in the form A cos (2t+ 0). Hint: Use the identity cos (x + y) =
cos x cos y — sin x sin y.

2
1
7.82. Find the orthogonal trajectories of the family of curves x — y = <?.
7.83. Find the orthogonal trajectories of the family of curves y = ce*.
2
2
7.84. Find the orthogonal trajectories of the family of curves x - y = ex.
2
7.85. Find the orthogonal trajectories of the family of curves x + y 2 = cy.
2
7.86. Find the orthogonal trajectories of the family of curves y = 4cx.
7.87. One hundred strands of bacteria are placed in a nutrient solution in which a plentiful supply of food is constantly
provided but space is limited. The competition for space will force the bacteria population to stabilize at 1000
strands. Under these conditions, the growth rate of bacteria is proportional to the product of the amount of bacteria
present in the culture with the difference between the maximum population the solution can sustain and the current
population. Estimate the amount of bacteria in the solution at any time t if it is known that there were 200 strands
of bacteria in the solution after seven hours.
7.88. A new product is to be test marketed by giving it free to 1000 people in a city of one million inhabitants, which is
assumed to remain constant for the period of the test. It is further assumed that the rate of product adoption will be
proportional to the number of people who have it with the number who do not. Estimate as a function of time the
number of people who will adopt the product if it is known that 3000 people have adopted the product after four
weeks.

7.89. A body of mass 1 slug is dropped with an initial velocity of 1 ft/sec and encounters a force due to air resistance
2
given exactly by -8v . Find the velocity at any time t.