Marcel B. Finan Arkansas Tech University c All Rights Reserved First Edition Fall 2010
1 A First Source of Diﬀerential Equations: Motion with Uniform Acceleration 4 2 Basic Concepts of Diﬀerential Equations 9
3 Existence and Uniqueness of Solutions to First Order Linear IVP 17 4 GraphicalSolution: Direction Field of y = f (t, y) 5 Analytical Solution: The Method of Integrating Factor 23 33
6 Existence and Uniqueness of Solutions to First Order Nonlinear IVP 44 7 Separable Diﬀerential Equations 8 Exact Diﬀerential Equations 9 Substitution Techniques: Bernoulli and Ricatti Equations 49 57 64
10 Numerical Solutions to ODEs: Euler’s Method and its Variants 70 11 Second OrderLinear Diﬀerential Equations: Existence and Uniqueness Results 78 12 The General Solution of Homogeneous Equations 83
13 Second Order Linear Homogeneous Equations with Constant Coeﬃcients 91 14 Characteristic Equations with Repeated Roots 15 Characteristic Equations with Complex Roots 96 101
16 The Structure of the General Solution of Linear Nonhomogeneous Equations 108
17 The Methodof Undetermined Coeﬃcients 18 The Method of Variation of Parameters 19 Review of Power Series 19.1 Power Series . . . . . . . . . . . . . . . . . . 19.2 Approximations by Taylor’s Polynomials . . 19.3 Taylor Series . . . . . . . . . . . . . . . . . . 19.4 Constructing New Taylor Series from Known 20 Series Solutions of Diﬀerential Equations 21 The Laplace Transform: Basic Deﬁnitions and Results 22Further Studies of Laplace Transform
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23 The Laplace Transform and the Method of Partial Fractions165 24 Laplace Transforms of Periodic Functions 25 Convolution Integrals 172 180
26 Modeling with First Order Linear Diﬀerential Equations: Mixing Problemsand Cooling Problems 185 27 Modeling with First Order Linear Diﬀerential Equations: Population Dynamics and Radioactive Decay 194 28 Modeling with First Order Nonlinear Equations: The Logistic Population Model 202 29 Modeling with a First Order Nonlinear Equations: OneDimensional Motion with Air Resistance 210 30 Modeling with a Homogeneous Second Order Linear Diﬀerential Equations: UnforcedMechanical Vibrations 224 31 Modeling with a Nonhomogeneous Second Order Linear Differential Equations: Forced Mechanical Vibrations 231
A First Source of Diﬀerential Equations: Motion with Uniform Acceleration
In many models, we will have equations involving the derivatives of a dependent variable y with respect to one or more independent variables and are interested in discoveringthis function y. Such equations are referred to as diﬀerential equations (abbreviated DE). They arise in many applications such as population growth, decay of radioactive substance, the motion of an object, electrical network, and many more models. In this section, we look at the motion of an object thrown upward from the top of a building. To this end, suppose that an object initially at height h0is thrown straight upward with velocity v0 . Let h(t) denote the object’s height from the ground, v(t) the object’s velocity, and a(t) the object’s acceleration at time t. Assume that air resistance is neglected. First, how does acceleration come about? Through the action of forces. Newton’s ﬁrst law of motion says that there is no change in the velocity of an object in the absence of a force.That is, the object is either at rest or moves at a constant velocity. His second law states that the presence of a force acts to produce a change in velocity, that is, an acceleration. It states that the net force is equal to the product of mass and acceleration. That is,
h Net force = m d 2 . dt
On the other hand, Newton’s law of gravity1 states that the net force between two objects is...