## Differential equations for engineers by Wei-Chau Xie

By Wei-Chau Xie

Similar civil engineering books

Partial differential equations in mechanics

This two-volume paintings makes a speciality of partial differential equations (PDEs) with vital purposes in mechanical and civil engineering, emphasizing mathematical correctness, research, and verification of options. The presentation consists of a dialogue of correct PDE purposes, its derivation, and the formula of constant boundary stipulations.

Practical Railway Engineering

This textbook is geared toward those that have to gather a extensive brush appreciation of all of the a variety of engineering capabilities which are eager about making plans, designing, developing, working and holding a railway approach. a sign is given of the diversities in those various disciplines among heavy rail, quick transit and lightweight rail operations.

Fundamentals of ground engineering

Basics of flooring Engineering is an unconventional research advisor that serves up the foremost ideas, theories, definitions, and analyses of geotechnical engineering in bite-sized items. This booklet includes brief―one or pages in step with topic―snippets of knowledge protecting the geotechnical engineering element of a standard undergraduate path in civil engineering in addition to a few subject matters for complicated classes.

Additional info for Differential equations for engineers

Sample text

Combining Cases 1 and 2, the solutions of the differential equation are y 2 + 2C x + C 2 = 0, y = ±x. 3 Exact Differential Equations and Integrating Factors Consider differential equations of the form M(x, y)dx + N(x, y)dy = 0, or M(x, y) dy =− , N(x, y) = 0, dx N(x, y) (1) ∂N ∂M and are continuous. Suppose the solution of equation (1) is ∂y ∂x u(x, y) = C, C = constant. Taking the differential yields where du = ∂u ∂u ∂u ∂u dx + dy = dC = 0 =⇒ dx + dy = 0. , the coefﬁcients of dx and dy in equations (1) and (2) are proportional ∂u ∂u ∂x = ∂y = μ(x, y) =⇒ ∂u = μM, M(x, y) ∂x N(x, y) ∂u = μN.

Equation (4) is variable separable, which can be solved easily by integration ln μ = ∂N 1 ∂M − dx =⇒ μ(x) = exp N ∂y ∂x ∂N 1 ∂M − dx . N ∂y ∂x (5) 40 2 ﬁrst-order and simple higher-order differential equations Note that, since only one integrating factor is sought, there is no need to include a constant of integration C. Interchanging M and N, and x and y in equation (5), one obtains an integrating factor for another special case μ( y) = exp ∂M 1 ∂N − dy . M ∂x ∂y function of y only Integrating Factors Consider the differential equation M(x, y)dx + N(x, y)dy = 0.

Dx 2v +1 2v +1 2v +1 Case 1. , (v −3)(v +2) = 0 y 3 = −2x or =⇒ v = −2 or v = 3, which gives y 3 = 3x. Case 2. v 2 −v −6 = 0, separating the variables gives 2v +1 v 2 −v −6 dv = 1 dx. x Variable separable Integrating both sides yields 2v +1 v 2 −v −6 dv = 1 dx + C. 2 method of transformation of variables 29 The ﬁrst integral can be evaluated using partial fractions (see pages 259–261 for a brief review on partial fractions) 2v +1 v 2 −v −6 = 2v +1 A B = + . (v −3)(v +2) v −3 v +2 Using the cover-up method, the coefﬁcients A and B can be easily determined A= 2v +1 v 2 −v −6 2v +1 v +2 dv = v=3 1 5 = 75 , B= 2v +1 v −3 v= − 2 = 35 , 7 3 + dv = 75 ln v −3 + 53 ln v +2 .