Finding Extremals using Euler Lagrange in Multivariable Calculus

We use the Euler Lagrange Equation to solve a functional with variational End points to find the Extremals of the Functional
F(y',y,x) = 1/2y^2 + yy' + y' + y
We find ourselves with a second order linear differential equation.
We find the homogeneous solution and the particular solution.
We find y(x) = Ax+b+x^2/2
Then using x=0 and x=1
We end up with a simultaneous equation to solve for our arbitrary constants
• A Gateaux Differential...
• Find the Gateaux Diffe...
#functionalanalysis
#functional
#calculus
#calculusofvariations
#calculusofvariation
#multivariablecalculus
#multivariable
#partial_differentiation
#partialderivatives
#partialdifferentiation