Suppose that f is a differentiable function of a single variable and F(x,y) is defined by F (x, y) = f (x2 − y).
a) Show that F satisfies the partial differential equation
b) Given that F(0,y) = sin y for all y, find a formula for F(x,y).
for a) I take LHS and subbed F for f (x2 − y).
which becomes
if F( 0, y ) = sin(y) thenF( 0, y) =f (0− y) = sin(y)
now the answer in the book is F(x,y) = sin ( y - x2 ) which is true F( 0, y ) = sin(y) but isn't the input of f dependant on x2 − y, can you just change that?
thanks
a) Show that F satisfies the partial differential equation
∂F/∂x + 2x•∂F/∂y = 0
b) Given that F(0,y) = sin y for all y, find a formula for F(x,y).
for a) I take LHS and subbed F for f (x2 − y).
∂/∂x(f (x2 − y)) + 2x•∂/∂y (f (x2 − y))
= 2x• f ' (x2 − y) - 2x• f ' (x2 − y) = 0
so LHS = RHS
for b) I'm a bit confusedso LHS = RHS
if F( 0, y ) = sin(y) thenF( 0, y) =f (0− y) = sin(y)
now the answer in the book is F(x,y) = sin ( y - x2 ) which is true F( 0, y ) = sin(y) but isn't the input of f dependant on x2 − y, can you just change that?
thanks
Last edited: