Problems
for Section 1.1.
1)
Verify by substitution that the given function is a solution of the given
differential equation:
|
a)
y(x) = x3 + 7 and y' = 3 x2 |
b)
y(x) = cos(2 x) and y'' + 4 y = 0 |
|
c)
y(x) = e-2x and y'' + 4 y' + 4 y = 0 |
d)
y(x) = 1/(1+x2) and y' + 2 x y2 = 0 |
2)
Find all values of r that make y(x) = erx a solution
of the differential equation
3 y' = 2 y
3)
First verify the given function satisfies the differential equation and then
find the constant C so that the function also satisfies the initial condition
|
a)
y(x) = C e-x, y' + y = 0 and y(0) = 2 |
b)
y(x) = ln(x+C), ey y' = 1 and y(0)=0 |
4)
The time rate of change of the velocity of a coasting motorboat is proportional
to the square of the velocity. Write down a differential equation describing
this.
5)
The slope of the graph of a function y=g(x) at a point (x,y) is proportional to
the sum of x and y. Write down a differential equation describing this.
6)
In a city having a fixed population of persons, the time rate of change of the
number of people who have heard a certain rumor is proportional to the number
of people that have not heard the rumor.
Write down a differential equation describing this.
7)
Guess one solution of the following differential equations:
|
a)
y'' = 0 |
b)
y' = y |