By Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith
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Additional resources for A Course in Higher Mathematics Volume II: Advanced Calculus
Lagrangian equations. An equation of the form: V = aPiW) + V»(in- (72) is called a Lagrangian equation, (p
55) We obtain the general solution in this form in the case of the equation with separable variables . The function co(x, y) on the left-hand side of (55), is called a solution of the differential equa tion (42). e. the solution of (42) is a function of x and y such that its total derivative with respect to x is zero, by virtue of (42). ,/ _ — U n > or, on replacing y' by f(x} y), inasmuch as y is a solution of (42) by hypothesis, we have: - ^ L + ^Lf^y) = 0. (56) 24 [7 ORDINARY DIFFERENTIAL EQUATIONS The function co(x, y) must satisfy this equation independently of the precise solution of (42) t h a t we have substituted in this function.
74) Substitution of this expression for x in equation (72) gives us an equation for y of the form: y = V3(p)0 + y>i(p). e. give the general solution of the Lagrangian equation in parametric form. On eliminating parameter p from (74) and (75), we get the ordinary equation for the general solution. e. e. the missing solutions must be straight lines, if they exist. e. the value of the constant a must be defined by the equation (p±(a) — a = 0. We give the geometrical interpretation of this last fact.