yJ = Ct ( * = l , 2 , . . , n ) , (45) with the essential proviso t h a t these equations are soluble with respect t o yv y2, . . , yn- Any equation of set (45) is called an integral of system (42), and n such integrals have to be found t o make u p the general solution of the system; thus it follows t h a t equations (45) must be soluble with respect to yv y2t .
59,) (6»i) The last two equations can be combined: (y-C)(y--±-x*-c}=0, giving the general solution of equation (59). Two integral curves pass through every point of the plane: the straight line (59x) and the parabola (592). Evidently (59x), y = C, gives a solution of (59) containing an arbitrary constant; this solution is not the general solution of (59), but only the general solution of the equation y' — 0. e. cannot be obtained from (52) with some particular value of constant C. Such a solution is called a singular solution of the equation.
The straight lines (70) form the family of tangents to it (Fig. 11). 9. Lagrangian equations. An equation of the form: V = aPiW) + V»(in- (72) is called a Lagrangian equation, (p
A Course of Higher Mathematics. Volume II by V. I. Smirnov and A. J. Lohwater (Auth.)