## Download PDF by Lev Elsgolts: Differential Equations And The Calculus Of Variations

By Lev Elsgolts

Один из лучших советских учебников - в переводе на английский.

The topic of this publication is the idea of differential equations and the calculus of diversifications. It relies on a direction of lectures which the writer brought for a host of years at the Physics division of the Lomonosov nation collage of Moscow.

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The constant of proportionality k, called the decay constant, is assumed known. 6) (the minus sign indicates a decrease in x as t increases, k Separating the variables and integrating, we get dx dt; In lxl-lnlxo I=- k(t-1 ) -x=-k > 0). 0 and then Let us also determine the half-life 't (that is, the time during which x0 /2 decays). Assuming t- 10 = 't, we get x 0 /2 = x 0e-kt, ln2 whence 't = T . Not only radioactive disintegration, but also any other monomolecular reaction is described on the basis of the mass action law, by the equation : = - kx.

Putting x =X+ I, y = Y + 2, we will have dY X-Y dX = X+Y. y The change of variables z =X or Y = zX leads to · the separable equation z + X ~ = 1- z (I + z) dz = dX dX I+ z' l-7z-z2 - ; In II-2z-z~ I= In I X 1- ~ X ' Inc, (l-2z-z:l) X 2 =c, X 2 -2XY -Y2 =c, x2 -2xy-y2 +2x+6y=c1 • 4. Linear Equations of the First Order A first-order linear differential equation is an equation that is linear in the unknown function and its derivative. 9). If f (x) 0, then the equation ( L 9) is called homogeneous linear.

22) might have been proved by a different method only if continuity of the function f (x, y) (without the Lipschitz condition) is assumed; however, continuity alone of the function f (x, y) is insufficient for proving the uniqueness of the solution. Note 2. The existence and uniqueness of the solution y= y (x) are proved only on the interval x0 -H~x~x0 +H; however, by taking the point (x0 + H, y (x0 +H)) for the initial point, it is possible, by repeating the reasoning, to extend the solution over I''· 52 I.

### Differential Equations And The Calculus Of Variations by Lev Elsgolts

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