Victoria University of Wellington
School of Mathematics, Statistics and Operations Research
MATH 244
2015
Differential Equations
Assignment 6: Solutions
1. Use the denition to nd the Laplace transform of the function:
x
0x1
f (x) = 2 x 1 < x 2 .
0
x>2
[3]
School of Mathematics, Statistics and Operations Research
MATH 244
2015
Differential Equations
Assignment 1 Solutions
1. For the second-order dierential equation y +9y = 0, show that y = c1 cos 3t+c2 sin 3t,
where c1 and c2 are constants, is always a solu
School of Mathematics, Statistics and Operations Research
MATH 244
2015
Differential Equations
Assignment 2 Solutions
1. For the following DEs, check they are linear rst-order by writing in standard form.
Then use an integrating factor to nd the general s
School of Mathematics, Statistics and Operations Research
MATH 244
Differential Equations
2015
Assignment 3 Solutions
1. Make this dierential equation exact by rst nding an integrating factor u:
[4]
(2y 2 + 3x) dx + (2xy) dy = 0.
Solution: Note that, sett
School of Mathematics, Statistics and Operations Research
MATH 244
Differential Equations
2015
Tutorial 3 Solutions
1. Make the dierential equation exact by rst nding an integrating factor u:
y(x + y + 1) dx + (x + 2y) dy = 0.
Solution: We know there will
School of Mathematics, Statistics and Operations Research
MATH 244
Differential Equations
2015
Tutorial 4 Solutions
1. The following procedure describes a programme for computing an approximate solution
to the IVP y = f (t, y), y(t0 ) = y0 by Eulers Metho
Victoria University of Wellington
School of Mathematics, Statistics and Operations Research
MATH 244
2015
Differential Equations
Assignment 7 solutions
1. Use the Second Translation Theorem to nd:
es
(a) Lcfw_x U(x 2)
(b) L1
(c) L1
s2 + 1
es
.
s2 + 2s + 2
School of Mathematics, Statistics and Operations Research
MATH 244
2015
Differential Equations
Assignment 4 Solutions
1. Use the Euler and improved Euler (Heun) methods to nd y(0.5) given y = 1 + y 2 ,
y(0) = 0 with t = 0.1. Solve this DE exactly and comp
Victoria University of Wellington
School of Mathematics, Statistics and Operations Research
MATH 244
2015
Ordinary Differential Equations
Assignment 8 Solutions
1. Find the eigenvalues and eigenfunctions for the BVP:
y + y = 0, y(0) = 0, y (/2) = 0. [5]
S
School of Mathematics, Statistics and Operations Research
MATH 244
2015
Differential Equations
Tutorial 5 Solutions
1. Check that the two standard solutions for the LHCC equation y + ay + b = 0 whose
auxiliary equation has a repeated root r = r1 are indep
EXAMINATIONS 2012
MATH 244
ORDINARY
DIFFERENTIAL EQUATIONS
Time Allowed: 3 hours
Instructions:
Question 1.
(a) [5 marks] Solve by separation of variables, the initial value problem
x2
dy
y =0,
dx
y (1) = 1 .
(b) [5 marks] State briey the main points of a
EXAMINATIONS 2011
MATH 244
DIFFERENTIAL EQUATIONS
Time Allowed: 3 hours
Instructions:
Question 1.
(a) [6 marks] Find the general solution to the dierential equation
(y + 1) ln x
dy
y
=
dx
x
(separate variables) .
Separating variables and integrating:
y+1
Differential Equations Formula Sheet
Trigonometric identities
sin2 x + cos2 x = 1
sin 2x = 2 sin x cos x
cos 2x = cos2 x sin2 x = 2 cos2 x 1 = 1 2 sin2 x
sec2 x = 1 + tan2 x
csc2 x = 1 + cot2 x
1
sin x cos y = [sin(x y ) + sin(x + y )]
2
1
sin x si