- #1
Somali_Physicist
- 117
- 13
I was trying to Extrapolate Eulers formula , after deriing the basic form I wanted to prove:
∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0
Here is my attempt but I get different answers:
J(y) = ∫abF(x,yx,y,yxx)dx
δ(ε) = J(y+εη(x))
y = yt+εη(x)
∂y/∂ε = η(x)
∂yx/∂ε = η⋅(x)
∂yxx/∂ε=η⋅⋅(x)(dots are derivatives with respect to x)
dδ(ε)/dε = ∫ab∂F/∂y ∂y/∂ε + ∂F/∂yx ∂yx/∂ε + ∂F/∂yxx ∂yxx/∂ε dx
∴ dδ(ε)/dε = ∫ab∂F/∂y η(x) + ∂F/∂yx η⋅(x) + ∂F/∂yxx η⋅⋅ (x)dx
Using integration by parts :
∫ab∂F/∂yx η⋅(x)dx = n(x)∂F/∂yx |ab-∫abd(∂F/∂yx)η(x)dx
n(x)∂F/∂yx |ab = 0
∴
∫ab∂F/∂yx η⋅(x)dx = -∫abd(∂F/∂yx)η(x)dx
Again use IBP:
∫ab∂F/∂yxx η⋅⋅ (x)dx = ∂F/∂yxxη⋅(x)|ab - ∫abd(∂F/∂yxx)/dxη⋅(x)dx
∫abd(∂F/∂yxx)/dxη⋅(x)dx = n(x)d(∂F/∂yxx)/dx |ab - ∫abd2(∂F/∂yxx)/dx2η(x)dx
n(x)d(∂F/∂yxx)/dx |ab = 0
hence
∫ab∂F/∂yxx η⋅⋅ (x)dx = ∂F/∂yxxη⋅(x)|ab + ∫abd2(∂F/∂yxx)/dx2η(x)dx
dδ(ε)/dε |ε=0 = dδ(ε)/dε = ∫ab∂F/∂yt η(x) -d(∂F/∂ytx)η(x)dx+ d2(∂F/∂ytxx)/dx2η(x)dx + ∂F/∂yxxη⋅(x)|ab = 0
Now this is where I am stuck , because I can intuitively derive the next step:
∂F/∂yxxη⋅(x)|ab =0 to give
(∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2)η(x) = 0
∴
as η(x) ≠ 0 (contradiction)
∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0Note assuming y(a,b) , yx(a,b) are fixed and differentiable on intervel
∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0
Here is my attempt but I get different answers:
J(y) = ∫abF(x,yx,y,yxx)dx
δ(ε) = J(y+εη(x))
y = yt+εη(x)
∂y/∂ε = η(x)
∂yx/∂ε = η⋅(x)
∂yxx/∂ε=η⋅⋅(x)(dots are derivatives with respect to x)
dδ(ε)/dε = ∫ab∂F/∂y ∂y/∂ε + ∂F/∂yx ∂yx/∂ε + ∂F/∂yxx ∂yxx/∂ε dx
∴ dδ(ε)/dε = ∫ab∂F/∂y η(x) + ∂F/∂yx η⋅(x) + ∂F/∂yxx η⋅⋅ (x)dx
Using integration by parts :
∫ab∂F/∂yx η⋅(x)dx = n(x)∂F/∂yx |ab-∫abd(∂F/∂yx)η(x)dx
n(x)∂F/∂yx |ab = 0
∴
∫ab∂F/∂yx η⋅(x)dx = -∫abd(∂F/∂yx)η(x)dx
Again use IBP:
∫ab∂F/∂yxx η⋅⋅ (x)dx = ∂F/∂yxxη⋅(x)|ab - ∫abd(∂F/∂yxx)/dxη⋅(x)dx
∫abd(∂F/∂yxx)/dxη⋅(x)dx = n(x)d(∂F/∂yxx)/dx |ab - ∫abd2(∂F/∂yxx)/dx2η(x)dx
n(x)d(∂F/∂yxx)/dx |ab = 0
hence
∫ab∂F/∂yxx η⋅⋅ (x)dx = ∂F/∂yxxη⋅(x)|ab + ∫abd2(∂F/∂yxx)/dx2η(x)dx
dδ(ε)/dε |ε=0 = dδ(ε)/dε = ∫ab∂F/∂yt η(x) -d(∂F/∂ytx)η(x)dx+ d2(∂F/∂ytxx)/dx2η(x)dx + ∂F/∂yxxη⋅(x)|ab = 0
Now this is where I am stuck , because I can intuitively derive the next step:
∂F/∂yxxη⋅(x)|ab =0 to give
(∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2)η(x) = 0
∴
as η(x) ≠ 0 (contradiction)
∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0Note assuming y(a,b) , yx(a,b) are fixed and differentiable on intervel