By R. Ansorge, W. Törnig
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Extra info for Numerische Behandlung nichtlinearer Integrodifferential- und Differentialgleichungen
01 to approximate the solution to the initial value problem y A0B ϭ 1 y¿ ϭ 2xy 2 , on the interval 0 Յ x Յ 2. ) Heat Exchange. There are basically two mechanisms by which a physical body exchanges heat with its environment. html, automates most of the differential equation algorithms discussed in this book. Technical Writing Exercises temperature and that of the environment; this is known as Newton’s law of cooling. However, heat transfer also occurs due to thermal radiation, which according to Stefan’s law of radiation is governed by the difference of the fourth powers of these temperatures.
Dx (a) A solution curve passes through the point A 1, p / 2 B . What is its slope at this point? (b) Argue that every solution curve is increasing for x Ͼ 1. (c) Show that the second derivative of every solution satisﬁes d 2y dx 2 ϭ 1 ϩ x cos y ϩ 1 2 sin 2y . (d) A solution curve passes through A 0, 0 B . Prove that this curve has a relative minimum at A 0, 0 B . 7. Consider the differential equation dp ϭ p A p Ϫ 1B A2 Ϫ pB dt for the population p (in thousands) of a certain species at time t. (a) Sketch the direction ﬁeld by using either a computer software package or the method of isoclines.
A) Show that f A x B ϭ x 2 is an explicit solution to x dy ϭ 2y dx on the interval A Ϫq, q B . (b) Show that f A x B ϭ e x Ϫ x is an explicit solution to dy ϩ y 2 ϭ e 2x ϩ A 1 Ϫ 2x B e x ϩ x 2 Ϫ 1 dx on the interval A Ϫq, q B . (c) Show that f A x B ϭ x 2 Ϫ x Ϫ1 is an explicit solution to x 2d 2y / dx 2 ϭ 2y on the interval A 0, q B . In Problems 3–8, determine whether the given function is a solution to the given differential equation. 3. x ϭ 2 cos t Ϫ 3 sin t , x – ϩ x ϭ 0 d 2y 4. y ϭ sin x ϩ x 2 , ϩ y ϭ x2 ϩ 2 dx 2 dx 5.