Intro Differential Equations.

MATH 2066 EL
Department of Mathematics and Computer Science
LAURENTIAN UNIVERSITY, SUDBURY
Deadline: Monday, 19 October 2015 at 9:30 a.m
October 4, 2015
EXERCISE 1. [Exact and Non Exact Equations; 24 Points = 6+6+6+6 ]
1. Show that each of the equations in Problems (a) through (b) is exact and solve the
given initial value problem.
(a)
dy
dx
= –
2xy + y2 + 1
x2 + 2xy
; y(-1) = 2.
(b) (yexy cos 2x – 2exy sin 2x + 2x) + (xexy cos 2x – 3)y0 = 0; y(?/4) = 0.
2. Show that the given equation is not exact, find an integrating factor and solve the
given equation
(x + 2) sin y + (x cos y)y0 = 0, x>0.
3. Show that the given equation is not exact but becomes exact when multiplied by the
given integrating factor. Then solve the equation.
?
sin y
y – 2e-x sin x
?
+
?
cos y + 2e-x cos x
y
?
y0 = 0, µ(x, y) = yex.
EXERCISE 2. [Bernouilli Equations; 24 Points = 7+17]
In each of Problem 1 through 2, find the solution of Bernouilli equations
1. y0 =
2
3t ln t
y + (ln t)2 1
py
, t>0, t 6= 1; y > 0.
2. y0 =
t
4(1 – t4)
y –
5t
4(1 + t2)2 y-3, y 6= 0; y(0) = 1.
1
EXERCISE 3. [Riccarti Equations; 18 Points= 8+10 ]
In each of Problem 1 through 2, solve the Riccarti equations satisfying the initial condition
given and where y1 is a particular solution.
1. y0 = (y – t)2 + 1, y1(t) = t; y(0) =
1
2
.
2. y0 = y2 –
y
x –
1
x2, x>0, y1(x) =
1
x
; y(1) = 2.
EXERCISE 4. [Phase line; 16 Points= 8+8 ]
Problems 1 through 2 involve equations of the form
dy
dx
= f(y). In each problem sketch the
graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one
asymptotically stable, unstable, or semistable. Draw the phase line in the ty-plane.
1.
dy
dx
= y(y2 – 3y + 2), y0 $ 0.
2.
dy
dx
= y2(1 – y2), -1 < y0 < 1.
EXERCISE 5. [18 Points= 4+5+4+5]
In each of Problem 1 through 4, find the general solution of the given di?erential equation
1. y00 – y0 – 12y = 0.
2. y000 – y00 – y0 + y = 0.
3. y00 – 2y0 + 5y = 0.
4. y000 + 6y00 + 12y0 + 8y = 0.
2

Intro Differential Equations.

MATH 2066 EL
Department of Mathematics and Computer Science
LAURENTIAN UNIVERSITY, SUDBURY
Deadline: Monday, 19 October 2015 at 9:30 a.m
October 4, 2015
EXERCISE 1. [Exact and Non Exact Equations; 24 Points = 6+6+6+6 ]
1. Show that each of the equations in Problems (a) through (b) is exact and solve the
given initial value problem.
(a)
dy
dx
= –
2xy + y2 + 1
x2 + 2xy
; y(-1) = 2.
(b) (yexy cos 2x – 2exy sin 2x + 2x) + (xexy cos 2x – 3)y0 = 0; y(?/4) = 0.
2. Show that the given equation is not exact, find an integrating factor and solve the
given equation
(x + 2) sin y + (x cos y)y0 = 0, x>0.
3. Show that the given equation is not exact but becomes exact when multiplied by the
given integrating factor. Then solve the equation.
?
sin y
y – 2e-x sin x
?
+
?
cos y + 2e-x cos x
y
?
y0 = 0, µ(x, y) = yex.
EXERCISE 2. [Bernouilli Equations; 24 Points = 7+17]
In each of Problem 1 through 2, find the solution of Bernouilli equations
1. y0 =
2
3t ln t
y + (ln t)2 1
py
, t>0, t 6= 1; y > 0.
2. y0 =
t
4(1 – t4)
y –
5t
4(1 + t2)2 y-3, y 6= 0; y(0) = 1.
1
EXERCISE 3. [Riccarti Equations; 18 Points= 8+10 ]
In each of Problem 1 through 2, solve the Riccarti equations satisfying the initial condition
given and where y1 is a particular solution.
1. y0 = (y – t)2 + 1, y1(t) = t; y(0) =
1
2
.
2. y0 = y2 –
y
x –
1
x2, x>0, y1(x) =
1
x
; y(1) = 2.
EXERCISE 4. [Phase line; 16 Points= 8+8 ]
Problems 1 through 2 involve equations of the form
dy
dx
= f(y). In each problem sketch the
graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one
asymptotically stable, unstable, or semistable. Draw the phase line in the ty-plane.
1.
dy
dx
= y(y2 – 3y + 2), y0 $ 0.
2.
dy
dx
= y2(1 – y2), -1 < y0 < 1.
EXERCISE 5. [18 Points= 4+5+4+5]
In each of Problem 1 through 4, find the general solution of the given di?erential equation
1. y00 – y0 – 12y = 0.
2. y000 – y00 – y0 + y = 0.
3. y00 – 2y0 + 5y = 0.
4. y000 + 6y00 + 12y0 + 8y = 0.
2