# New PDF release: Applied Hydro - and Aeromechanics. Based on Lectures by L By Oskar Karl Gustav Tietjens

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Extra resources for Applied Hydro - and Aeromechanics. Based on Lectures by L Prandtl. (Transl. by Jacob Pieter Den Hartog.)

Sample text

A first-order linear equation has the form u + p(t)u = q(t). 52) becomes dt (uef P(t)dt) = q(t)ef p(t)dt Now, both sides can be integrated to determine u. We illustrate this procedure with an example. 8 Find an expression for the solution to the initial value problem u'+2tu=2f, u(0)=3. 1. Dimensional Analysis, Scaling, and Differential Equations 38 The integrating factor is exp(f 2t dt) = exp(t2). Multiplying both sides of the equation by the integrating factor makes the left side a total derivative, or (uet2 )' = 2Viet2 .

F. u"+tu'2=0 in. u"+W2u=sinfit, g. u" - 3u' - 4u = 2sint. n. u" + w2u = Cos wt. 3 Differential Equations 41 2. ) Show that the change of variables y = u%t transforms the equation into a separable equation for y. 3. (Particle dynamics) An object of mass m is shot vertically upward with initial velocity vo. The force of gravity, of magnitude mg, acts on the object, as well as a force due to air resistance that is proportional to the square of the velocity. a) Use Newton's second law to write down the governing initial value problem.

Integrating from 0 to t (while changing the dummy variable of integration to s) gives /u(x) t u (t)et2 - u(0) = 2 fo es2ds. Solving for u gives = e-t2 (3 + f 2= 3e-t2 + J 220 As is often the case, the integrals in this example cannot be performed easily, if at all, and we must write the solution in terms of integrals with variable limits. Some first-order equations can be solved by substitution. For example, Bernoulli equations are differential equations having the form u' + p(t)u = q(t)u'2. The transformation of dependent variables w = u1-n turns a Bernoulli equation into a linear equation for w = w(t).