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By zalman rubinstein

ISBN-10: 1483230988

ISBN-13: 9781483230986

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Additional info for A Course in Ordinary and Partial Differential Equations

Sample text

The solution is obtained as follows: y' = z, y" = zy' z. , d~y = Z = ey + d and 1 dy - = x + c2 e +c y Ci (y - log(ey + cO). Solving for y, we have logci-\og[e-Cl(x+C2)-ll. y = To satisfy the initial conditions, Cl = e ~ c l C 2 - 1 = c 1 e" C l C 2 , which implies c*! = 0. For q=0, j ; = log( \-c2 ) -x] and it also follows that c2 = — 1. The solution y = — log(l — x) is defined for — oo < x < 1. 3. EXISTENCE THEOREMS 41 EXERCISES 1. Show that y(x) = x2 is not a solution on [0, 1] of any equation y' = F(x,y) with \F(x,y)\ < 1 for 0 < x < 1, 0

Is a uniform Cauchy sequence and hence converges uniformly to a continuous function y{x) defined on the interval [x° — h, x° + It]. Furthermore '1/n(*) -y°- ffiu yi/n(t)) dt\\ = II fV /B '(x) -/(*, /'"'(O)) dt < - (x - x°). 32 ORDINARY DIFFERENTIAL EQUATIONS Since f(x, y ,n(x)) tends uniformly to f(x, y(x)) as n -> oo, we finally obtain y(x)-y°- ff(t,y(t))dt = 0. x)=f(x,y(x)), y(x°)=y° and y(x) is a continuously differentiable function. Theorem 3: Under the hypothesis of Theorem 2, the solution of the differential equation (19) satisfying the initial condition y(x°) = y° is unique.

28 ORDINARY DIFFERENTIAL EQUATIONS for all y, z e D. The norm || || is defined by \\y\\ = max{|j i |, i= 1 , 2 , . . , « } . Note that if y = y(x) then ||j>|| = |>>|| (x). For every point (x°, y°) e D there exists an h > 0, such that the problem y' =f(x,y),y(x°) = y° has a continuously differentiable solution for all x that satisfies \x — x°\ (), fli>o. l Since D is open and (x°, j>°) e Z>, such /^ and a exist. Now by the continuity of / i n Z), there exists a positive number ζ^ such that ||/|| < Q1 in Dx.

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A Course in Ordinary and Partial Differential Equations by zalman rubinstein


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