Implicit function on functions composed of itself

chy1013m1
Messages
14
Reaction score
0
Suppose F(x, y) is C1. F(0, 0) = 0. What conditions on F will guarantee that the equation F(F(x, y), y) = 0 can be solved for y as a C1 function of x near (0, 0) ?

would it simply be dF/dy not equal 0 ?
 
Physics news on Phys.org
Use the implicit function theorem. Seems you are on the right track, but F is a composition of F and y, so you would have to use the chain rule to right out dF/dy properly.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Can You Apply the Implicit Function Theorem Correctly?
Replies
6
Views
1K
Proof given ##x < y < z## and a twice differentiable function
Replies
8
Views
1K
Implicit function theorem part 2
Replies
1
Views
1K
Chain Rule Confusion (Euler-Lagrange Equation)
Replies
5
Views
972
Relationship between autonomous system and related single equation
Replies
3
Views
2K
Applying the implicit function theorem to a system of equations
Replies
10
Views
2K
How should I show that ## B ## is given by the solution of this?
Replies
2
Views
1K
Why is this valid? Continuity of Functions Problem
Replies
3
Views
1K
Can this function still be a constant function?
Replies
11
Views
1K
How should I use the Jacobi equation to determine the nature of this?
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top