home2_09F - Department of Mathematical Sciences University...

Department of Mathematical
Sciences
University of Delaware
Prof. T. Angell
September 16, 2009
Mathematics 351
Exercise Sheet 2
Exercise 6 : Consider the initial value problem for the nonhomogeneous linear equation
dy
dx
= − y + x , y (0) = 4 .
(a) Show that y ( x ) = x − 1 is a solution of the di ff erential equation .
(b) Find a one parameter family of solutions of the di ff erential equation. Determine the parameter
value corresponding to the given initial condition.
(c) Show that the graphs of all solutions of the di ff erential equation are asymptotic to the line y = x − 1
as x → ∞ .
Exercise 7 : Find all solutions of the di ff erential equation
t
dx
dt
− 4 x = t
6
e
t
, t > 0 .
Exercise 8 : Show that the functions y
1
( t ) ≡ 0, and y
2
( t ) = ( t − t
o
)
3
are both solutions of the initial
value problem y = 3 y
2 / 3
, y ( t
o
) = 0 on −∞ < t < ∞ . Graph both solutions for the successive values of
t
o
= − 1 , 0 , 1.
Exercise 9 : An equation of the form y + p ( t ) y = q ( t ) y n , is called a Bernoulli equation.
(a) Solve Bernoulli’s equation when n = 0 and when n = 1.
(b) Recalling that if n = 0 , 1 then the substitution v = y
1 − n
reduces Bernoulli’s equation to a linear
equation, solve t
2
y + 2