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 = 3x^{5}/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 = lncos(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" = 12x^{3};
(ii) y" = 4e^{2x};
(iii) y" = sec^{2}(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 

yaxis 
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 = Ce^{2x}  1/2
Explicit solution with y = 0 when x = 0 is y = 1/2 e^{2x}  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´ = y1;
(ii) y" = 2y´ = 4y+2;
(iii) y" = 8y^{3 }y´ = 16y^{7};
(iv) y" = 3y^{4}y´=^{ }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, semistable 
iv 
y>0 
y<0 
y<0 
y>0 
about xaxis 
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  y^{2}/10  8/5 = (y2)(y8)/10
equilibrium solutions: y = 2 which is unstable
y = 8 which is stable
(ii) y´ = y  y^{2}/10  5/2 = (y5)^{2}/10
equilibrium solution : y = 5 which is semistable
(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}.