MATH2270

FW00


Solutions to Assignment 1


Note the use of the following: pi - no greek symbol available; <> for 'not equal to'.

1. (a)

(i) Analytic family of solutions:

Explicit solution with y = 5 when x = 2 is y = 3x5/5 - 71/5


(ii) Analytic family of solutions:

Explicit solution with y = 2 when x = 0 is y = -e-2x + 3


(iii) Analytic family of solutions:

Explicit solution with y = 1 when x = 0 is y = -ln|cos(x)| + 1


(iv) Analytic family of solutions:

Explicit solution with y = 1 when x = 1 is y = -(x+1)-1 + 3/2.


(b) We need the second derivatives of the functions in order to determine the concavity. These are:

(i) y" = 12x3;

(ii) y" = -4e-2x;

(iii) y" = sec2(x);

(iv) y" = -2(x+1)-3;


question

monotonicity

concavity

symmetries

singularities


increasing

decreasing

up

down



i

all x


x>0

x<0

about origin

none

ii

all x



all x

none

none

iii*

0<x<pi/2

-pi/2<x<0

all x<>(n+1/2)pi


y-axis

pi/2, - pi/2

iv

all x<>-1


x<-1

x>-1

none#

x=-1

* the more general answer for this part is given below where n is an integer

npi<x<(n+1/2)pi; (n+1/2)pi<x<(n+1)pi; singularities at (n+1/2)pi;

#the solution is symmetric for rotations through 180 degrees about the point (-1, 0)


(c) See MAPLE output



2. (a) (i) Analytic family of solutions: or y = 1 + Ce-x

Explicit solution with y = 5 when x = 0 is y = 1 + 4e-x.


(ii) Analytic family of solutions: or y = Ce2x - 1/2

Explicit solution with y = 0 when x = 0 is y = 1/2 e2x - 1/2

(iii) Analytic family of solutions:

Explicit solution with y = 3 when x = 1 is


(iv) Analytic family of solutions:

which is defined for x > -c.

Explicit solution with y = -1 when x = -2 is y = -[4x + 9]1/4

(b) We need the second derivatives of the functions in order to determine the concavity. These are:

(i) y" = -y´ = y-1;

(ii) y" = 2y´ = 4y+2;

(iii) y" = 8y3 y´ = 16y7;

(iv) y" = -3y-4y´= -3y-7;


question

monotonicity

concavity

symmetries

singularities

equilibrium


increasing

decreasing

up

down



solutions

i

y<1

y>1

y>1

y<1

none*

none

y=1,stable

ii

y>-1/2

y<-1/2

y>-1/2

y<-1/2

none#

none

y=-1/2, unstable

iii

all y


y>0

y<0

about origin

none

y=0, semi-stable

iv

y>0

y<0

y<0

y>0

about x-axis

y=0

none

* solution is symmetric about the line y = 1; # solution is symmetric about the line y = -1/2.


(c) See MAPLE output



3. (a)

(i) y´ = y - y2/10 - 8/5 = -(y-2)(y-8)/10

equilibrium solutions: y = 2 which is unstable

y = 8 which is stable

(ii) y´ = y - y2/10 - 5/2 = -(y-5)2/10

equilibrium solution : y = 5 which is semi-stable


(b), (c) see MAPLE output



4. The differential equation for radioactive decay is the exponential differential equation dy/dt = -ay with a = ln(2)/60.5 = 0.011457 where t is in units of minutes. The analytic family of solutions is

y = Ce-0.011457t.