MAT 216 - WINTER 2014
Homework 1
Ryan Szypowski
Due January 16, 2014
1. Determine the general solution to the following dierential equations:
(a) y = (1 + y 2 ) sin t
(b) y = (1 y 2 )ex
(c) y + y = sin(2t)
(d) y + 2 y =
t
cos t
t2
2. Determine the solutio
MAT 216 - WINTER 2014
Homework 7 Solutions
Ryan Szypowski
Due March 13, 2014
1. Use the Laplace transform to solve the following initial value problems:
(a) y y 2y = 30et cos(2t) + 10et sin(2t), y(0) = 6, y (0) = 16
Solution: Taking the Laplace transform
MAT 216 - WINTER 2014
Homework 3 Solutions
Ryan Szypowski
Due January 30, 2014
1. Sketch the phase line diagram for the autonomous dierential equation
y = y(y 5)(y 10)
and describe the behavior as t if y(0) = 6.
Solution: The phase line would have stable
MAT 216 - WINTER 2014
Homework 6
Ryan Szypowski
Due February 27, 2014
1. Determine the homogeneous solution and the form of the particular solution given the
roots r and right hand sides g(t) below.
(a) r = 1, 3, 4, 4 2i, 4 2i,
(b) r = 1, 2, 2, 2,
g(t) =
MAT 216 - WINTER 2014
Homework 5
Ryan Szypowski
Due February 20, 2014
1. Determine the general solution to the following second order ODEs.
(a) y + 4y + 4y = t2 e2t
(b) y 2y + y =
et
1+t2
(c) x2 y 3xy + 4y = x2 ln x, given that cfw_x2 , x2 ln x forms a fu
MAT 216 - WINTER 2014
Homework 6 Solutions
Ryan Szypowski
Due February 27, 2014
1. Determine the homogeneous solution and the form of the particular solution given the
roots r and right hand sides g(t) below.
(a) r = 1, 3, 4, 4 2i, 4 2i, g(t) = 19t2 e4t
S
MAT 216 - WINTER 2014
Homework 1 Solutions
Ryan Szypowski
Due January 16, 2014
1. Determine the general solution to the following dierential equations:
(a) y = (1 + y 2 ) sin t
Solution: This equation is separable, and so we begin with
dy
= sin t dt.
1 +
MAT 216 - WINTER 2014
Homework 4 Solutions
Ryan Szypowski
Due February 13, 2014
1. Determine the general solution to the following second order ODEs.
(a) y 6y + 34y = 0
Solution: Consider the characteristic polynomial
r2 6r + 34 = 0.
This has roots of r =
MAT 216 - WINTER 2014
Homework 4
Ryan Szypowski
Due February 13, 2014
1. Determine the general solution to the following second order ODEs.
(a) y 6y + 34y = 0
(b) y 14y + 49y = 2809 cos(2t)
(c) y 5y + 6y = 36t
2. Find the solution to the following initial
MAT 216 - WINTER 2014
Homework 2 Solutions
Ryan Szypowski
Due January 23, 2014
1. Determine the general solution to the following dierential equations:
2
y
(a) The homogeneous equation y = 2 x+y
Solution: We can recognize that this equation is homogeneous
MAT 216 - WINTER 2014
Homework 2
Ryan Szypowski
Due January 23, 2014
1. Determine the general solution to the following dierential equations:
2
y
x+y
(a) The homogeneous equation y = 2
(b) The Bernoulli equation (x + 1)(y + y 2 ) = y
2. Determine the solu
MAT 216 - WINTER 2014
Homework 3
Ryan Szypowski
Due January 30, 2014
1. Sketch the phase line diagram for the autonomous dierential equation
y = y(y 5)(y 10)
and describe the behavior as t if y(0) = 6.
2. Build a function f (y) such that the dierential eq
MAT 216 - WINTER 2014
Homework 7
Ryan Szypowski
Due March 13, 2014
1. Use the Laplace transform to solve the following initial value problems:
(a) y y 2y = 30et cos(2t) + 10et sin(2t),
(b) y 6y + 12y 8y = 0,
(c) y 5y + 6y = g(t),
y(0) = 2,
y(0) = 1,
y(0)