Download e-book for kindle: A Course of Higher Mathematics. Volume II by V. I. Smirnov and A. J. Lohwater (Auth.)

By V. I. Smirnov and A. J. Lohwater (Auth.)

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Extra resources for A Course of Higher Mathematics. Volume II

Example text

If x = O is the point of intersection of a chord with OXy the square of the radius of the corresponding circle is (1 — O2) (Fig. 13), so that the equation of the family is: (x-C)2 + y2 = l - O2. -!. e. we obtain an ellipse with semi-axes |^2 and 1, with the coordinate axes as axes of symmetry. I t is obvious from the figure that this ellipse does not touch all the circles of the family. 11. Equations quadratic in y', We consider in more detail, from the point of view of singular solutions, differential equations that are quadratic in t/': ® (*.

I t is only among these curves that singular solutions of equation (84) can be found. e. ,' + P(M)=0. The latter equation expresses the fact that (84) has a multiple root with respect to y\ 1. e. y'2 + xy' — y = 0 (86) takes the form # 2 /4 + y = 0, and the parabola y — — # 2 /4 is a singular solution of the Clairaut equation written. 2. In the case of the equation y'2 -f 2xtf + y = 0 (86) gives y = x2. This parabola does not satisfy the equation written, so that the latter has no singular solution whatever.

0 For the case when the beam is supported rigidly at the end x = 0 and is subjected to concentrated loading at the other end x = I, we have [16]: ■? / = a- 2/ - a ) —■ J /l-fc2x2(2/-a02 dx. 58 ORDINARY DIFFERENTIAL EQUATIONS [18 The integral written is not expressible in terms of elementary functions. The curve corresponding to equation (41) is called an elastic curve. 3. We take the equation: & yy" = (y — %y')2> both sides of which are homogeneous functions of yt y\ y".

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