# Download A Course of Higher Mathematics. Volume II by V. I. Smirnov and A. J. Lohwater (Auth.) PDF By V. I. Smirnov and A. J. Lohwater (Auth.)

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Extra info for A Course of Higher Mathematics. Volume II

Example text

2/onl)We thus obtain the solution satisfying conditions (5). If the righthand side of equation (2) is a many-valued function, there will be several solutions of equation (7) corresponding to initial conditions (5). e. not obtainable from (6) for any values of constants Cs, is called a singular solution of the equation. The remarks made in  in connection with first order equations must be borne in mind as regards the concepts of general solution 42 [13 ORDINABY DIFFERENTIAL EQUATIONS and singular solutions.

5 1 9 . . , which corresponds to a maximum for \y\. 54 ORDINARY DIFFERENTIAL EQUATIONS [17 Maximum deflection thus occurs towards the end L and not at the centre, its value being: ^ — l/lx-if. 348- 360j57/ I8QJM-• 17. Lowering the order of a differential equation. We notice a number of particular cases in which the order of an equation can be lowered. 1. Let the function y and a certain number of consecutive derivatives of y:y',y", . . , y**""1*, be excluded from the equation, which has t h e form: &(xyyW,y(k+1\..

Y^n~x\ a single definite solution of equation (2) corresponds to initial conditions (5). On varying the constants y0, y&, . . , yon~^ in the initial conditions, we obtain an infinite set of solutions, or more accurately, a family of solutions, depending on n arbitrary constants. ,Cn). (6) Such a solution of equation (2), containing n arbitrary constants, is called the general solution of (2). ,Cn) = 0. (7) On assigning definite values to constants Cv C2, . . , Cn, we obtain particular solutions of the equation.