- #1

lep11

- 380

- 7

Member advised that the homework template is not optional

Let ##h(u,v)=f(u+v,u-v)## and ##f_{xx}=f_{yy}## for every ##(x,y)\in\mathbb{R}^2##. In addition, ##f\in{C^2}.## Show that ##f(x,y)=U(x+y)+V(x-y)##.

I think applying the Taylor theorem would be useful.

##f(x,y)=f(x+h_1,y+h_2)-\left(\frac{\partial{f(x,y)}}{\partial{x}}h_1+\frac{\partial{f(x,y)}}{\partial{y}}h_2\right)-\frac 1 2\left(\frac{\partial^2{f(x,y)}}{\partial^2{x}}h_1^2+\frac{\partial^2{f(x,y)}}{\partial{x}\partial{y}}h_1h_2+\frac{\partial^2{f(x,y)}}{\partial^2{y}}h_2^2\right)-R(h_1,h_2)##

How would one proceed?

An alternative, maybe more elegant way would be to denote ##x+y=\epsilon##, ##x-y=\mu## and solve the hyperbolic partial differential equation ##\frac{\partial^2{f}}{\partial{\epsilon}\partial{\mu}}=0##, but we haven't actually covered them in class yet.

##\frac{\partial^2{f}}{\partial{\epsilon}\partial{\mu}}=0## ⇔ ##\frac{\partial{f}}{\partial{\epsilon}}=C(\epsilon)## etc.

Let ##h(u,v)=f(u+v,u-v)## and ##f_{xx}=f_{yy}## for every ##(x,y)\in\mathbb{R}^2##. In addition, ##f\in{C^2}.## Show that ##f(x,y)=U(x+y)+V(x-y)##.

I think applying the Taylor theorem would be useful.

##f(x,y)=f(x+h_1,y+h_2)-\left(\frac{\partial{f(x,y)}}{\partial{x}}h_1+\frac{\partial{f(x,y)}}{\partial{y}}h_2\right)-\frac 1 2\left(\frac{\partial^2{f(x,y)}}{\partial^2{x}}h_1^2+\frac{\partial^2{f(x,y)}}{\partial{x}\partial{y}}h_1h_2+\frac{\partial^2{f(x,y)}}{\partial^2{y}}h_2^2\right)-R(h_1,h_2)##

How would one proceed?

An alternative, maybe more elegant way would be to denote ##x+y=\epsilon##, ##x-y=\mu## and solve the hyperbolic partial differential equation ##\frac{\partial^2{f}}{\partial{\epsilon}\partial{\mu}}=0##, but we haven't actually covered them in class yet.

##\frac{\partial^2{f}}{\partial{\epsilon}\partial{\mu}}=0## ⇔ ##\frac{\partial{f}}{\partial{\epsilon}}=C(\epsilon)## etc.

Last edited by a moderator: