function pvextrcond_2_restr_integral
clc,clear all
disp('Se rezolva problema variationala de extrem conditionat cu 2 restrictii de tip integral')
syms x y z Dy D2y Dz D2z lambda1 lambda2 % variabilele
F=Dy*Dz; % integrandul
x1=0;y1=0;z1=0;x2=1;y2=0;z2=1; % conditiile la extremitati
F1=y; J1=1;F2=z; J2=0; % restrictiile de tip integral
disp('Datele initiale:')
fprintf(['Functia integrand ' 'F(x,y,y'',z,z'')=%s\n'],char(F))
disp('Conditiile la extremitatea de stanga:')
fprintf('y(%d)=%d;   z(%d)=%d\n',x1,y1,x1,z1)
disp('Conditiile la extremitatea de dreapta:')
fprintf('y(%d)=%d;   z(%d)=%d\n',x2,y2,x2,z2)
disp('Conditiile de tip integral:')
fprintf('Int(%s,''x'',%d,%d)=%d\n',char(F1),x1,x2,J1)
fprintf('Int(%s,''x'',%d,%d)=%d\n',char(F2),x1,x2,J2)
L=F+lambda1*F1+lambda2*F2; % functia lui Lagrange
fprintf('L(x,y,y'',lambda1,lambda2)=%s\n',char(L))
dLdy=diff(L,y); dLdy1=diff(L,Dy); d_dLdy1_dx=diff(dLdy1,x);
d_dLdy1_dy=diff(dLdy1,y); d_dLdy1_dy1=diff(dLdy1,Dy); 
d_dLdy1_dz=diff(dLdy1,z); d_dLdy1_dz1=diff(dLdy1,Dz);
dLy1dx=d_dLdy1_dx+d_dLdy1_dy*Dy+d_dLdy1_dy1*D2y+d_dLdy1_dz*Dz+d_dLdy1_dz1*D2z;
dLdz=diff(L,z);dLdz1=diff(L,Dz); d_dLdz1_dx=diff(dLdz1,x);
d_dLdz1_dy=diff(dLdz1,y); d_dLdz1_dy1=diff(dLdz1,Dy);
d_dLdz1_dz=diff(dLdz1,z); d_dLdz1_dz1=diff(dLdz1,Dz);
dLz1dx=d_dLdz1_dx+d_dLdz1_dy*Dy+d_dLdz1_dy1*D2y+d_dLdz1_dz*Dz+d_dLdz1_dz1*D2z;
EulerYL=simple(dLdy-dLy1dx);
EulerZL=simple(dLdz-dLz1dx);
dEuYL=[char(EulerYL) '=0']; % ecuatia intai a SEEL asociat functionalei lui Lagrange
dEuZL=[char(EulerZL) '=0']; % ecuatia intai a SEEL asociat functionalei lui Lagrange
fprintf('EEL pentru problema asociata de extrem neconditionat:\n%s\n%s\n',dEuYL,dEuZL)
SolL=dsolve(dEuYL,dEuZL,'x'); % se rexolva prin metoda exacta SEEL
if length(SolL)~=1
  error('Nu sunt solutii sau sunt mai multe!');
else
  SolYL=SolL.y; SolZL=SolL.z;
  disp('Solutia generala a SEEL:')
  fprintf('y(x)=%s\nz(x)=%s\n',char(SolYL),char(SolZL))
end
dydx=diff(SolYL,x); dzdx=diff(SolZL,x); % dy/dx
F1_yz=subs(F1,{y,Dy,z,Dz},{SolYL,dydx,SolZL,dzdx});
F2_yz=subs(F2,{y,Dy,z,Dz},{SolYL,dydx,SolZL,dzdx});
intF1=vpa(int(F1_yz,x,x1,x2),14); % se calculeaza integrala
intF2=vpa(int(F2_yz,x,x1,x2),14);
disp('Membrii stangi ai restrictiilor de tip integral:')
disp(char(intF1))
disp(char(intF2))
SolYLleft=vpa(subs(SolYL,x,x1),14);
SolYLright=vpa(subs(SolYL,x,x2),14);
LeftYL=[char(SolYLleft) '=' char(sym(y1))];
RightYL=[char(SolYLright) '=' char(sym(y2))];
SolZLleft=vpa(subs(SolZL,x,x1),14);
SolZLright=vpa(subs(SolZL,x,x2),14);
LeftZL=[char(SolZLleft) '=' char(sym(z1))];
RightZL=[char(SolZLright) '=' char(sym(z2))];
intF1J1=[char(intF1) '=' char(sym(J1))];
intF2J2=[char(intF2) '=' char(sym(J2))];
disp('Sistemul de ecuatii in raport cu C1,C2,C3,C4,lambda1,lambda2')
fprintf('%s\n',LeftYL,RightYL,intF1J1,LeftZL,RightZL,intF2J2)
ConL=solve(LeftYL,RightYL,intF1J1,LeftZL,RightZL,intF2J2,'C2,C3,lambda2,C1,C4,lambda1');
C1=vpa(ConL.C1,14); C2=vpa(ConL.C2,14);C3=vpa(ConL.C3,14); C4=vpa(ConL.C4,14);
lambda1=vpa(ConL.lambda1,14); lambda2=vpa(ConL.lambda2,14); % multiplicatorii Lagrange
Sol141Y=vpa(eval(SolYL),14); Sol141Z=vpa(eval(SolZL),14); % solutia analitica
disp('Componentele extremalei:')
fprintf('y(x)=%s\nz(x)=%s\n',char(Sol141Y),char(Sol141Z))
disp('Multiplicatorii Lagrange')
fprintf('lambda1=%s\nlambda2=%s\n',char(lambda1),char(lambda2))
F141=subs(F,{y,z,Dy,Dz},{Sol141Y,Sol141Z,diff(Sol141Y,x),diff(Sol141Z,x)});
J141c=eval(int(F141,x,x1,x2))
xpl=linspace(x1,x2); y141=subs(Sol141Y,x,xpl); z141=subs(Sol141Z,x,xpl);
plot(xpl,y141,'-.b',xpl,z141,'--g') 
set(get(gcf,'CurrentAxes'),'FontName','Times New Roman','FontSize',11)
title('\bfProblema variationala de extrem conditionat cu restrictii de tip integral') 
xlabel('\itx') 
ylabel('\ity\rm(\itx\rm)')
legend('y(x)','z(x)',0)
grid on
box on
figure
plot3(xpl,y141,z141,'-r') % se deseneaza graficul 3D
set(get(gcf,'CurrentAxes'),'FontName','Times New Roman','FontSize',11)
title('\bfGraficul 3D')
xlabel('\itx')
ylabel('\ity\rm(\itx\rm)')
zlabel('\itz\rm(\itx\rm)')
view(205,30) % selectia punctului de vizualizare
grid on
box on
end