Calculus of Variations solvedproblems - cuni.cz

1 Calculus of Variationssolved problemsPavel PyrihJune 4, 2012( public domain )Acknowledgement. The following problems were solved using my own procedurein a program Maple V, release 5. All possible errors are my Solving the Euler equationTheorem.(Euler) Supposef(x, y, y ) has continuous partial derivatives of thesecond order on the interval [a, b]. If a functionalF(y) = baf(x, y, y )dxattainsa weak relative extrema aty0, theny0is a solution of the following equation f y ddx( f y )= is called theEuler the Euler equation find the extremals for the following functional ba12xy(x) + ( xy(x))2 denote auxiliary functionff(x,y(x), xy(x)) = 12xy(x) + ( xy(x))2in the formf(x, y, z) = 12x y+z2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

2 We compute partial derivatives yf(x,y(x), xy(x)) = 12xand zf(x,y(x), xy(x)) = 2 ( xy)and 2 x zf(x,y(x), xy(x)) = 2 ( 2 x2y(x))and finally obtain the Euler equation for our functional12x 2 ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) =x3+C1x+ the Euler equation find the extremals for the following functional ba3x+ xy(x) denote auxiliary functionff(x,y(x), xy(x)) = 3x+ xy(x)in the formf(x, y, z) = 3x+ zSubstituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional. We compute partial derivatives yf(x,y(x), xy(x)) = 0and zf(x,y(x), xy(x)) =121 xyand 2 x zf(x,y(x), xy(x)) = 14 2 x2y(x)( xy(x))(3/2)and finally obtain the Euler equation for our functional14 2 x2y(x)( xy(x))(3/2)= 0We solve it using various tools and obtain the solutiony(x) =C1x+ the Euler equation find the extremals for the following functional bax+ y(x)2+ 3 ( xy(x)) denote auxiliary functionff(x,y(x), xy(x)) =x+ y(x)2+ 3 ( xy(x))in the formf(x, y, z) =x+y2+ 3zSubstituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

3 We compute partial derivatives yf(x,y(x), xy(x)) = 2yand zf(x,y(x), xy(x)) = 3and 2 x zf(x,y(x), xy(x)) = 0and finally obtain the Euler equation for our functional2 y(x) = 0We solve it using various tools and obtain the solutiony(x) = the Euler equation find the extremals for the following functional bay(x)2 denote auxiliary functionff(x,y(x), xy(x)) = y(x)2in the formf(x, y, z) =y2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional. We compute partial derivatives yf(x,y(x), xy(x)) = 2yand zf(x,y(x), xy(x)) = 0and 2 x zf(x,y(x), xy(x)) = 0and finally obtain the Euler equation for our functional2 y(x) = 0We solve it using various tools and obtain the solutiony(x) = the Euler equation find the extremals for the following functional bay(x)2+x2( xy(x)) denote auxiliary functionff(x,y(x), xy(x)) = y(x)2+x2( xy(x))in the formf(x, y, z) =y2+x2zSubstituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

4 We compute partial derivatives yf(x,y(x), xy(x)) = 2yand zf(x,y(x), xy(x)) =x2and 2 x zf(x,y(x), xy(x)) = 2xand finally obtain the Euler equation for our functional2 y(x) 2x= 0We solve it using various tools and obtain the solutiony(x) = the Euler equation find the extremals for the following functional bay(x) +x( xy(x)) denote auxiliary functionff(x,y(x), xy(x)) = y(x) +x( xy(x))in the formf(x, y, z) =y+x zSubstituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional. We compute partial derivatives yf(x,y(x), xy(x)) = 1and zf(x,y(x), xy(x)) =xand 2 x zf(x,y(x), xy(x)) = 1and finally obtain the Euler equation for our functional0 = 0We solve it using various tools and obtain the solutiony(x) = Y(x) the Euler equation find the extremals for the following functional ba xy(x) denote auxiliary functionff(x,y(x), xy(x)) = xy(x)in the formf(x, y, z) =zSubstituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

5 We compute partial derivatives yf(x,y(x), xy(x)) = 0and zf(x,y(x), xy(x)) = 1and 2 x zf(x,y(x), xy(x)) = 0and finally obtain the Euler equation for our functional0 = 0We solve it using various tools and obtain the solutiony(x) = Y(x) the Euler equation find the extremals for the following functional bay(x)2 ( xy(x))2 denote auxiliary functionff(x,y(x), xy(x)) = y(x)2 ( xy(x))2in the formf(x, y, z) =y2 z2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional. We compute partial derivatives yf(x,y(x), xy(x)) = 2yand zf(x,y(x), xy(x)) = 2 ( xy)and 2 x zf(x,y(x), xy(x)) = 2 ( 2 x2y(x))and finally obtain the Euler equation for our functional2 y(x) + 2 ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) =C1cos(x) +C2sin(x) the Euler equation find the extremals for the following functional ba 1 + ( xy(x))2 denote auxiliary functionff(x,y(x), xy(x)) = 1 + ( xy(x))2in the formf(x, y, z) = 1 +z2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

6 We compute partial derivatives yf(x,y(x), xy(x)) = 0and zf(x,y(x), xy(x)) = xy 1 + ( xy)2and 2 x zf(x,y(x), xy(x)) = ( xy(x))2( 2 x2y(x))(1 + ( xy(x))2)(3/2)+ 2 x2y(x) 1 + ( xy(x))2and finally obtain the Euler equation for our functional( xy(x))2( 2 x2y(x))(1 + ( xy(x))2)(3/2) 2 x2y(x) 1 + ( xy(x))2= 0We solve it using various tools and obtain the solutiony(x) =C1x+ the Euler equation find the extremals for the following functional bax( xy(x)) + ( xy(x))2 denote auxiliary functionff(x,y(x), xy(x)) =x( xy(x)) + ( xy(x))2in the formf(x, y, z) =x z+z2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional. We compute partial derivatives yf(x,y(x), xy(x)) = 0and zf(x,y(x), xy(x)) =x+ 2 ( xy)and 2 x zf(x,y(x), xy(x)) = 1 + 2 ( 2 x2y(x))and finally obtain the Euler equation for our functional 1 2 ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) = 14x2+C1x+ the Euler equation find the extremals for the following functional ba(1 +x) ( xy(x))2 denote auxiliary functionff(x,y(x), xy(x)) = (1 +x) ( xy(x))2in the formf(x, y, z) = (1 +x)z2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

7 We compute partial derivatives yf(x,y(x), xy(x)) = 0and zf(x,y(x), xy(x)) = 2 (1 +x) ( xy)and 2 x zf(x,y(x), xy(x)) = 2 ( xy(x)) + 2 (1 +x) ( 2 x2y(x))and finally obtain the Euler equation for our functional 2 ( xy(x)) 2 (1 +x) ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) =C1+C2ln(1 +x) the Euler equation find the extremals for the following functional ba2 y(x)ex+ y(x)2+ ( xy(x))2 denote auxiliary functionff(x,y(x), xy(x)) = 2 y(x)ex+ y(x)2+ ( xy(x))2in the formf(x, y, z) = 2y ex+y2+z2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional. We compute partial derivatives yf(x,y(x), xy(x)) = 2ex+ 2yand zf(x,y(x), xy(x)) = 2 ( xy)and 2 x zf(x,y(x), xy(x)) = 2 ( 2 x2y(x))and finally obtain the Euler equation for our functional2ex+ 2 y(x) 2 ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) = (12cosh(x)2+12cosh(x) sinh(x) +12x) sinh(x)+ ( 12cosh(x) sinh(x) +12x 12cosh(x)2) cosh(x) +C1sinh(x) +C2cosh(x) the Euler equation find the extremals for the following functional ba2 y(x) + ( xy(x))2 denote auxiliary functionff(x,y(x), xy(x)) = 2 y(x) + ( xy(x))2in the formf(x, y, z) = 2y+z2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

8 We compute partial derivatives yf(x,y(x), xy(x)) = 2and zf(x,y(x), xy(x)) = 2 ( xy)and 2 x zf(x,y(x), xy(x)) = 2 ( 2 x2y(x))and finally obtain the Euler equation for our functional2 2 ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) =12x2+C1x+ the Euler equation find the extremals for the following functional ba2 ( xy(x))3 denote auxiliary functionff(x,y(x), xy(x)) = 2 ( xy(x))3in the formf(x, y, z) = 2z3 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional. We compute partial derivatives yf(x,y(x), xy(x)) = 0and zf(x,y(x), xy(x)) = 6 ( xy)2and 2 x zf(x,y(x), xy(x)) = 12 ( xy(x)) ( 2 x2y(x))and finally obtain the Euler equation for our functional 12 ( xy(x)) ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) = the Euler equation find the extremals for the following functional ba4x( xy(x)) + ( xy(x))2 denote auxiliary functionff(x,y(x), xy(x)) = 4x( xy(x)) + ( xy(x))2in the formf(x, y, z) = 4x z+z2 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

9 We compute partial derivatives yf(x,y(x), xy(x)) = 0and zf(x,y(x), xy(x)) = 4x+ 2 ( xy)and 2 x zf(x,y(x), xy(x)) = 4 + 2 ( 2 x2y(x))and finally obtain the Euler equation for our functional 4 2 ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) = x2+C1x+ the Euler equation find the extremals for the following functional ba7 ( xy(x))3 denote auxiliary functionff(x,y(x), xy(x)) = 7 ( xy(x))3in the formf(x, y, z) = 7z3 Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional. We compute partial derivatives yf(x,y(x), xy(x)) = 0and zf(x,y(x), xy(x)) = 21 ( xy)2and 2 x zf(x,y(x), xy(x)) = 42 ( xy(x)) ( 2 x2y(x))and finally obtain the Euler equation for our functional 42 ( xy(x)) ( 2 x2y(x)) = 0We solve it using various tools and obtain the solutiony(x) = the Euler equation find the extremals for the following functional ba y(x) (1 + ( xy(x))2) denote auxiliary functionff(x,y(x), xy(x)) = y(x) (1 + ( xy(x))2)in the formf(x, y, z) = y(1 +z2)Substituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.

10 We compute partial derivatives yf(x,y(x), xy(x)) =121 + ( xy)2 y(1 + ( xy)2)and zf(x,y(x), xy(x)) =y( xy) y(1 + ( xy)2)and 2 x zf(x,y(x), xy(x)) = 12y(x) ( xy(x)) (( xy(x)) (1 + ( xy(x))2) + 2 y(x) ( xy(x)) ( 2 x2y(x)))(y(x) (1 + ( xy(x))2))(3/2)+( xy(x))2 y(x) (1 + ( xy(x))2)+y(x) ( 2 x2y(x)) y(x) (1 + ( xy(x))2)and finally obtain the Euler equation for our functional121 y(x)+12y(x) ( xy(x)) (( xy(x)) (1 + ( xy(x))2) + 2 y(x) ( xy(x)) ( 2 x2y(x)))(y(x) (1 + ( xy(x))2))(3/2) ( xy(x))2 y(x) (1 + ( xy(x))2) y(x) ( 2 x2y(x)) y(x) (1 + ( xy(x))2)= 018We solve it using various tools and obtain the solutiony(x) =144 +C12x2+ 2C12xC2+ the Euler equation find the extremals for the following functional baxy(x) ( xy(x)) denote auxiliary functionff(x,y(x), xy(x)) =xy(x) ( xy(x))in the formf(x, y, z) =x y zSubstituting wex,y(x) andy (x) forx,yandzwe obtain the integrand in thegiven functional.