By J. David Logan (auth.)
This concise and updated textbook is designed for a standard sophomore path in differential equations. It treats the fundamental rules, versions, and resolution equipment in a person pleasant structure that's available to engineers, scientists, economists, and arithmetic majors. It emphasizes analytical, graphical, and numerical suggestions, and it presents the instruments wanted by way of scholars to proceed to the following point in employing the easy methods to extra complex difficulties. there's a robust connection to functions with motivations in mechanics and warmth move, circuits, biology, economics, chemical reactors, and different parts. Exceeding the 1st variation via over 100 pages, this re-creation has a wide raise within the variety of labored examples and perform workouts, and it maintains to supply templates for MATLAB and Maple instructions and codes which are important in differential equations. pattern exam questions are integrated for college students and teachers. suggestions of the various workouts are contained in an appendix. in addition, the textual content includes a new, hassle-free bankruptcy on structures of differential equations, either linear and nonlinear, that introduces key principles with out matrix research. next chapters deal with platforms in a extra formal means. in brief, the themes contain: * First-order equations: separable, linear, independent, and bifurcation phenomena; * Second-order linear homogeneous and non-homogeneous equations; * Laplace transforms; and * Linear and nonlinear structures, and part airplane homes.
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Extra info for A First Course in Differential Equations
Generalize the method of Exercise 8 by devising a method to solve u′ = au + q(t), where a is any constant and q is a given function. In fact, show that t u(t) = Ceat + eat e−as q(s)ds. 0 Using the fundamental theorem of calculus, verify that this function does solve u′ = au + q(t). 4 Mathematical Models 29 10. Use the chain rule and the fundamental theorem of calculus to compute the derivative of erf(sin t). 11. The Dawson function is defined by the expression t 2 2 es ds. D(t) = e−t 0 Find the differential equation for D(t).
The fundamental questions are: (a) is there a solution curve passing through the given point, (b) is the curve the only one, and (c) what is the interval (α, β) on which the solution exists. 2. (Uniqueness) If there is a solution, is the solution unique? That is, is it the only solution? This is the question of uniqueness. 3. (Interval of Existence) For which times t does the solution to the initial value problem exist? Obtaining resolution of these theoretical issues is an interesting and worthwhile endeavor, and it is the subject of advanced courses and books on differential equations.
Show that this equation can have a “decaying-oscillation” solution of the form x(t) = e−λt cos ωt for some λ and ω. Hint: Substitute into the differential equation; show that the decay constant λ and frequency ω can be determined in terms of the known parameters m, c, and k. 3. A car of mass m is moving at speed V when it has to brake. The brakes apply a constant force F until the car comes to rest. How long does it take the car to stop? How far does the car go before stopping? Now, with specific data, compare the time and distance it takes to stop if you are going 30 mph versus 35 mph.