Mathematics for Economists
Calculus of variations The boundary conditions are 0 = ^1 6 +c 1 +c 2 1 = ^8 3 + 2 c 1 +c 2 that is c 1 +c 2 = 1 6 ,^2 c^1 +c^2 ...
Calculus of variations Example Find the extremal of the functional Z 3 1 ( 3 tx)xdt, x( 1 )= 1 ,x( 3 )= 4 1 2. The kernel functi ...
Calculus of variations Example Find the extremal solutions of Z 1 0 exp(x)+tx dt ,x( (^0) )= 0 ,x( (^1) )=α. The EulerñLagrange ...
Calculus of variations Example Find the extremal of the functional Z 2 π 0 x^2 (t)x^2 (t)dt, x( 0 )=x( 2 π)= 1. The kernel func ...
Calculus of variations Example Find Z 1 0 r 1 + x 2 dt!min x( 0 )= 0 ,x( 1 )= 1. The kernel function is F t,x,x = r 1 ...
Calculus of variations c= x r 1 + x 2 c^2 1 + x 2 = x 2 1 c^2 x 2 =c^2 x=C)x=Ct+B. Using the initial ...
Calculus of variations We show that the kernel is convex. d dx p 1 +x^2 = p x x^2 + 1 d^2 dx^2 p 1 +x^2 = p x^2 + 1 xpxx (^2) + ...
Calculus of variations Example Find the extremal Z 1 0 y y^0 2 dx,y( 0 )= 1 ,y( 1 )=^3 p 4 The Euler-Lagrange equation is y^ ...
Calculus of variations It is a second order non-linear equation. In this case one can try to substituteu=y^0 and assume thaty^0 ...
Calculus of variations Using this 2 yy^00 + y^0 2 = 0 2 ydu dx + y^0 2 = 0 2 y(x) d dyu(y(x)) d dxy(x)+u (^2) (x) = 0 2 y( ...
Calculus of variations First we solve the equation 2 yu(y) d dy u(y)+u^2 (y)= 0. Ifu(y)=0 theny^0 (x)=0 theny(x)is a constant wh ...
Calculus of variations This is again an equation with separable variables c q jyjdy = dx c 1 jyj^3 /^2 = x+c 2 y = c 33 q (x+c 2 ...
Calculus of variations Example Find the extremal of Ze 1 x y^0 2 +yy^0 dx,y( 1 )= 0 ,y(e)= 1. The EulerñLagrange equation is ...
Calculus of variations Again introduceu=y^0 .(It is always working whenyis missing.) Then 0 = u+xu^0. ^1 x dx =^1 u du lnjxj = l ...
Calculus of variations Example Find the extremal solution of Zb a y+(y (^0) )^3 3 dx The EulerñLagrange equation is 1 = d dx y^ ...
Calculus of variations Example 2 π ZT 0 x(t) r 1 + x 2 dt!min x( 0 )=a,x(T)=b The kernel is F t,x,x =x r 1 + x 2 ...
Calculus of variations r 1 + x 2 = r 1 + x 2 xx 0 (^) r 1 + x 2!^0 xx 1 + x 2 r 1 + x 2 = r 1 + ...
Calculus of variations 1 + x 2 = x 2 +xx+ x 4 +xx x 2 x x (^2) x 1 + x 2 = 1 + x 4 + 2 ...
Calculus of variations It is a non-linear equation. Assume thatx^0 depends only onxand not on t.Letu=x^0 ,x^00 =dtdu^0 =du/dxdx ...
Calculus of variations x 2 =x 2 C^2 1 Writing back to the EulerñLagrange xx x^2 C^2 1 1 = 0 xx =x 2 C^2 x = x ...
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