By J. David Logan

This article is designed for a standard post-calculus direction in simple differential equations. it's a short, one-semester therapy of the fundamental principles, versions, and answer equipment. The ebook, which serves in its place to present texts for teachers who wish extra concise insurance, emphasizes graphical, analytical, and numerical methods, and is written with transparent language in a straight forward structure. It offers scholars with the instruments to proceed directly to the subsequent point in using differential equations to difficulties in engineering, technology, and utilized mathematics.

The subject matters include:
* separable and linear first-order equations;
* self sustaining equations;
* moment order linear homogeneous and nonhomogeneous equations;
* Laplace transforms;
* linear and nonlinear structures within the section plane.

Many routines are supplied, as well as examples from engineering, ecology, physics, economics, and different parts. An extended part at the required linear algebra is gifted, and an appendix includes templates of Maple and MATLAB instructions and courses that are important in differential equations.

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Extra resources for A First Course in Differential Equations (Undergraduate Texts in Mathematics)

Sample text

This is a second-order diﬀerential equation for the unknown location or position x = x(t) of the particle. The force F may depend on time t, position x = x(t), or velocity x = x (t). This DE is called the equation of motion or the dynamical equation for the system. For second-order diﬀerential equations we impose two initial conditions, x(0) = x0 and x (0) = v0 , which ﬁx the initial position and initial velocity of the particle, respectively. We expect that if the initial position and velocity are known, then the equation of motion should determine the state for all times t > 0.

4◦ F. Estimate the time of death. 4. M. and your furnace then fails. M. M.? 5. The temperature T (t) of an exothermic, chemically reacting sample placed in a furnace is governed by the initial value problem T = −k(T − Te ) + qe−θ/T , T (0) = T0 , where the term qe−θ/T is the rate heat is generated by the reaction. What are the dimensions of all the constants (k, Te , q, T0 , and θ) in the problem? Scale time by k −1 and temperature by Te to obtain the dimensionless model dψ = −(ψ − 1) + ae−b/ψ , dτ ψ(0) = γ, 48 1.

Find the solution to the initial value problem u = e√t , u(1) = 0, in terms of an integral. Graph the solution on the interval [1, 4] by using numerical integration to calculate values of the integral. 6. The diﬀerential equation u = 3u + e−t can be converted into a pure time equation for a new dependent variable y using the transformation u = ye3t . Find the pure time equation for y, solve it, and then determine the general solution u of the original equation. 7. Generalize the method of Exercise 6 by devising an algorithm to solve u = au + q(t), where a is any constant and q is a given function.