Exercises

1.
Find the general solution of the equation y′′ -6y′ + 5y = 0.BITMAPSETAnswer0.2214in0.205in0ina1

2.
Find the general solution of the equation x2y′′ -3xy′ - 6y = 0.BITMAPSETAnswer0.2214in0.205in0ina2

3.
Find the general solution of the equation x2y′ = xy + 3y2.BITMAPSETAnswer0.2214in0.205in0ina3

4.
Solve the initial-value problem y′ + y = 2, y(0) = 0.BITMAPSETAnswer0.2214in0.205in0ina4

5.
Solve the initial-value problem ${\frac{{dy}}{{dt}}}$ - y - 3x = 0, -5y + ${\frac{{dx}}{{dt}}}$ - 3x = 0, y(0) = 1, x(0) = - 1.

BITMAPSETAnswer0.2214in0.205in0ina5

6.
The model ${\frac{{dP}}{{dt}}}$ = kP(M - P) describes a population P limited by a maximum sustainable population M, where k is a growth-rate constant. Predict the population ten years from now, assuming a current population of 500, if the maximum sustainable population is 2000 and the population ten years ago was 300.BITMAPSETAnswer0.2214in0.205in0ina6

7.
The height of a hanging cable satisfies the differential equation y′′ = α$\sqrt{{1+(y^{\prime })^{2}}}$. Find a solution for this differential equation with α = 1, assuming y(0) = 1 and y′(0) = 0.BITMAPSETAnswer0.2214in0.205in0ina7

8.
Newton's law of cooling states that the rate of change in the temperature of an object is given by ${\frac{{dT}}{{dt}}}$ = k(T - R), where k is a constant that depends on how well insulated the object is, T is the temperature of the object, and R is room temperature. A cup of coffee is initially 160o; 10 minutes later, it is 120o. Assuming the room temperature is a constant 70o, give a formula for the temperature at any time t. What will the temperature of the coffee be after 20 minutes?BITMAPSETAnswer0.2214in0.205in0ina8

9.
Coyotes (C), rabbits (R), and vegetation (V) are about the only living things on a small, isolated island. The coyotes rely on rabbits for food, and the rabbits eat the vegetation. The biomass is governed by the differential equations BITMAPSETAnswer0.2214in0.205in0ina9
$\displaystyle {\dfrac{{dC}}{{dt}}}$ = -0.2C + 0.0004CR  
$\displaystyle {\dfrac{{dR}}{{dt}}}$ = -0.01CR + .001RV  
$\displaystyle {\dfrac{{dV}}{{dt}}}$ = -0.001RV + .001V(1000 - V)  

where t is measured in years. Assume an initial population of 100 coyotes, 1000 rabbits, and 1000 units of vegetation. Predict the values of C, R, and V over the next 5 years. (Choose Numeric.)

10.
Let v and u denote the horizontal and vertical components of velocity, respectively, of a golf ball in flight, and let $\left(\vphantom{ x,y}\right.$x, y$\left.\vphantom{ x,y}\right)$ denote its position. Define the following constants. BITMAPSETAnswer0.2214in0.205in0ina10
w = $\displaystyle {\frac{{1.62}}{{16}}}$, weight of the ball in pounds  
c = $\displaystyle {\frac{{1}}{{200}}}$w, drag term  
g = 32, acceleration due to gravity in ft/s2  
θ = $\displaystyle {\frac{{\pi }}{{8}}}$, angle of the club head  
k = c, lift due to backspin  
z = 150, club head velocity in ft/s  

Solve the following system of differential equations numerically, then plot (x, y) parametrically to view the flight of a golf ball.


$\displaystyle {\frac{{dv}}{{dt}}}$ = - $\displaystyle {\frac{{gcv}}{w}}$  
$\displaystyle {\frac{{du}}{{dt}}}$ = - $\displaystyle {\frac{g}{w}}$$\displaystyle \left(\vphantom{ w+cu-kv}\right.$w + cu - kv$\displaystyle \left.\vphantom{ w+cu-kv}\right)$  
$\displaystyle {\frac{{dx}}{{dt}}}$ = v  
$\displaystyle {\frac{{dy}}{{dt}}}$ = u  
v(0) = z cos2θ  
u(0) = z cosθsinθ  
x(0) = 0  
y(0) = 0  

11.
Solve the partial differential equation Dxu + 3Dxyu = xy2. Verify the solution by defining F1(x) and F2(x) to be generic functions and substituting the suggested solution back into the partial differential equation.BITMAPSETAnswer0.2214in0.205in0ina11