Homework 1
For the following two problems, verify by substitution that the given function
is a solution of the given dierential equation. Throughout these problems,
primes denote dierentiation with respect to x.
1. y00 = 9y; y1 = e3x, y2 = e 3x
2. x2 y00 + xy0 y = ln x; y1 = x ln x, y2 = x1 ln x
3. Substitute y = erx into the dierential equation
3y0 = 2y
to determine all values of the constant r for which y = erx is a solution of
the equation.
For the following three problems, rst verify that y(x) satises the given
dierential equation. Then determine the value of the constant C so that y(x)
satises the given initial condition.
4. y0 = x y; y(x) = Ce x + x 1, y(0) = 10
dy
5. x dx
+ 3y = 2x5 ; y(x) = 14 x5 + Cx 3 , y(2) = 1
6. y0 + y tan x = cos x; y(x) = (x + C ) cos x, y() = 0
For the following two problems, a function y = g(x) is described by some
geometric properties of ites graph. Write a dierential equation of the form
dy=dx = f (x; y) having the function g as its solution.
7. The slope of the graph of g at the point (x; y) is the sum of x and y.
8. Every straight line normal to the graph of g passes through the point
(0; 1). Can you guess what the graph of such a function g might look like?
9. Suppose a population P of rodents satises the dierential equation
dP = kP 2 :
dt
Initially, there are P (0) = 2 rodents, and their number is increasing at the
rate of dP=dt = 1 rodent per month when there are P = 10 rodents. How
long will it take for this population to grow to one hundred rodents? To
one thousand?
10. Suppose the velocity of a motorboat coasting in water satises the dierential equation
dv = kv2 :
dt
The initial speed of the motorboat is v(0) = 10 meters per second (m=s),
and v is decreasing at the rate of 1 m=s2 when v = 5 m=s. How long does
it take for the velocity of the boat to decrease to 1 m=s? To 1/10 m=s?
When does the boat come to a stop?
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