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(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.