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

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