Course Notes for AMath 250

J. Wainwright1

Department of Applied Mathematics

University of Waterloo

March 9, 2010

1

c J. Wainwright, April 2003

Contents

1 First Order Diﬀerential Equations

1.1 DEs and Mechanics . . . . . . . . . . . . . . . . . . . .

1.1.1 Newton’s Second Law of Motion . . . . . . . . .

1.1.2 Dimensions of physical quantities . . . . . . . .

1.1.3 Newton’s Law of Gravitation . . . . . . . . . .

1.2 Mathematical aspects of ﬁrst order DEs . . . . . . . .

1.2.1 Types of ﬁrst order DEs . . . . . . . . . . . . .

1.2.2 Solving separable DEs . . . . . . . . . . . . . .

1.2.3 Solving linear DEs . . . . . . . . . . . . . . . .

1.2.4 Qualitative sketches of families of solutions . . .

1.2.5 First order linear DEs with constant coeﬃcient

1.2.6 An important special case . . . . . . . . . . . .

1.2.7 A common error . . . . . . . . . . . . . . . . .

1.2.8 Initial value problems . . . . . . . . . . . . . . .

1.3 Other applications of ﬁrst order DEs . . . . . . . . . .

1.3.1 Mixing problems . . . . . . . . . . . . . . . . .

1.3.2 Population growth . . . . . . . . . . . . . . . .

1.3.3 Epidemics . . . . . . . . . . . . . . . . . . . . .

1.3.4 Cooling problems . . . . . . . . . . . . . . . . .

1.3.5 Pursuit problems . . . . . . . . . . . . . . . . .

1.3.6 Electrical circuits . . . . . . . . . . . . . . . . .

2 Dimensional Analysis

2.1 Writing physical relations in dimensionless form . . . .

2.1.1 Characteristic scales and dimensionless variables

2.1.2 The mixing tank DE . . . . . . . . . . . . . . .

2.1.3 The sky-diver DE . . . . . . . . . . . . . . . . .

2.2 Deducing physical relations using dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1 A motivating example . . . . . . . . . . . . . .

2.2.2 Complete sets of dimensionless variables . . . .

2.2.3 The Buckingham Pi Theorem . . . . . . . . . .

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References: Borelli, R.L. and Coleman, C.S., 1987, Diﬀerential Equations: A Modeling Approach, PrenticeHall. Goldberg, J. and Potter, M.C., 1998, Diﬀerential Equations: A Systems Approach, PrenticeHall. Simmons, G.F., 1972, Diﬀerential Equations – with applications and historical notes, McGrawHill. Reiss, E.L., Callegari, A.J., Ahluwalia, D.S., 1976, Ordinary diﬀerential equations with applications, Holt, Rinehart & Winston. Brauer, F. and Nohel, J.A., 1967, Ordinary Diﬀerential Equations, W.A. Benjamin. Braun, M., 1983, Diﬀerential Equations and their Applications, Springer-Verlag. Boyce, W.E. and diPrima, R.C., 1997, Elementary Diﬀerential Equations and Boundary Value Problems, 6th edition, J