In[1]:= Needs["VariationalMethods`"]
L = z'[x]^2 - 2*(x*Cos[x] + Sin[x])*z'[x] - y[x]^2 +
l[x]*(z[x] - y'[x] - x*Sin[x]);
eq1 = EulerEquations[L, y[x], x];
eq2 = EulerEquations[L, z[x], x];
eq3 = {-x*Sin[x] + z[x] - y'[x] == 0, y[0] == 1, y[Pi/2] == 0,
z[0] == 0, z[Pi/2] == -1};
In[6]:= s = DSolve[{eq1, eq2, eq3}, {l, y, z}, x]
Out[6]= {{l ->
Function[{x}, -(1/8) E^(-(\[Pi]/2) -
x) (-2 E^(\[Pi]/2) \[Pi] + 2 E^\[Pi] \[Pi] + 2 E^(2 x) \[Pi] +
2 E^(\[Pi]/2 + 2 x) \[Pi] - 4 E^(\[Pi]/2 + x) Cos[x] +
2 E^x \[Pi] Cos[x] + 2 E^(\[Pi] + x) \[Pi] Cos[x] +
4 E^(\[Pi]/2 + x) Cos[x] Cos[2 x] -
16 E^(\[Pi]/2 + x) Sin[x] - 2 E^x \[Pi] Sin[x] -
4 E^(\[Pi]/2 + x) \[Pi] Sin[x] +
2 E^(\[Pi] + x) \[Pi] Sin[x] - 2 E^(\[Pi]/2) Cos[x] Sin[x] +
2 E^(\[Pi]/2 + 2 x) Cos[x] Sin[x] +
4 E^(\[Pi]/2 + x) Cos[x] Sin[x]^2 + E^(\[Pi]/2) Sin[2 x] -
E^(\[Pi]/2 + 2 x) Sin[2 x] +
2 E^(\[Pi]/2 + x) Sin[x] Sin[2 x])],
y -> Function[{x}, -(1/16) E^(-(\[Pi]/2) -
x) (2 E^(\[Pi]/2) \[Pi] - 2 E^\[Pi] \[Pi] + 2 E^(2 x) \[Pi] +
2 E^(\[Pi]/2 + 2 x) \[Pi] - 16 E^(\[Pi]/2 + x) Cos[x] -
2 E^x \[Pi] Cos[x] - 4 E^(\[Pi]/2 + x) \[Pi] Cos[x] +
2 E^(\[Pi] + x) \[Pi] Cos[x] - 4 E^(\[Pi]/2 + x) Sin[x] -
2 E^x \[Pi] Sin[x] - 2 E^(\[Pi] + x) \[Pi] Sin[x] +
2 E^(\[Pi]/2) Cos[x] Sin[x] +
2 E^(\[Pi]/2 + 2 x) Cos[x] Sin[x] +
4 E^(\[Pi]/2 + x) Cos[x]^2 Sin[x] -
4 E^(\[Pi]/2 + x) Cos[2 x] Sin[x] - E^(\[Pi]/2) Sin[2 x] -
E^(\[Pi]/2 + 2 x) Sin[2 x] +
2 E^(\[Pi]/2 + x) Cos[x] Sin[2 x])],
z -> Function[{x}, -(1/16) E^(-(\[Pi]/2) -
x) (-2 E^(\[Pi]/2) \[Pi] + 2 E^\[Pi] \[Pi] + 2 E^(2 x) \[Pi] +
2 E^(\[Pi]/2 + 2 x) \[Pi] + 4 E^(\[Pi]/2 + x) Cos[x] -
2 E^x \[Pi] Cos[x] - 2 E^(\[Pi] + x) \[Pi] Cos[x] -
4 E^(\[Pi]/2 + x) Cos[x] Cos[2 x] +
16 E^(\[Pi]/2 + x) Sin[x] + 2 E^x \[Pi] Sin[x] +
4 E^(\[Pi]/2 + x) \[Pi] Sin[x] -
2 E^(\[Pi] + x) \[Pi] Sin[x] - 16 E^(\[Pi]/2 + x) x Sin[x] -
2 E^(\[Pi]/2) Cos[x] Sin[x] +
2 E^(\[Pi]/2 + 2 x) Cos[x] Sin[x] -
4 E^(\[Pi]/2 + x) Cos[x] Sin[x]^2 + E^(\[Pi]/2) Sin[2 x] -
E^(\[Pi]/2 + 2 x) Sin[2 x] -
2 E^(\[Pi]/2 + x) Sin[x] Sin[2 x])]}}