An interesting functional equation The Next CEO of Stack OverflowIterative Functional EquationA functional equationTrivial funcional equationFunctional equation $f(ax)=bf(x)$Solving a particular Functional Differential EquationFunctional equation $xspace f(x^2) = f(x)$Question regarding the Cauchy functional equationMean Value theorem functional equationRequest for interesting Functional Equations, of a specific typeFunctional Equation $f(x+f(x))=x+f(x)$
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An interesting functional equation
The Next CEO of Stack OverflowIterative Functional EquationA functional equationTrivial funcional equationFunctional equation $f(ax)=bf(x)$Solving a particular Functional Differential EquationFunctional equation $xspace f(x^2) = f(x)$Question regarding the Cauchy functional equationMean Value theorem functional equationRequest for interesting Functional Equations, of a specific typeFunctional Equation $f(x+f(x))=x+f(x)$
$begingroup$
$$frac1-fBig(fracxx+(1-x)f(x)Big)1-f(x) = 1-x(1-x)fracf'(x)f(x)$$
Now, we know that $f(x)=c$ and $f(x) = fraca+bx1-x$ are two solutions. How can I get other solutions or to prove that these are all the solutions? Thanks!
functional-equations
edited Mar 27 at 19:04
Maria Mazur
49k1360122
asked Mar 27 at 18:57
ftorftor
113
$endgroup$
add a comment |
$begingroup$
$$frac1-fBig(fracxx+(1-x)f(x)Big)1-f(x) = 1-x(1-x)fracf'(x)f(x)$$
Now, we know that $f(x)=c$ and $f(x) = fraca+bx1-x$ are two solutions. How can I get other solutions or to prove that these are all the solutions? Thanks!
functional-equations
edited Mar 27 at 19:04
Maria Mazur
49k1360122
asked Mar 27 at 18:57
ftorftor
113
$endgroup$
$begingroup$
I would substitute $$t=fracxx+(1-x)f(x)$$
$endgroup$
– Dr. Sonnhard Graubner
Mar 27 at 19:02
add a comment |
$begingroup$
$$frac1-fBig(fracxx+(1-x)f(x)Big)1-f(x) = 1-x(1-x)fracf'(x)f(x)$$
Now, we know that $f(x)=c$ and $f(x) = fraca+bx1-x$ are two solutions. How can I get other solutions or to prove that these are all the solutions? Thanks!
functional-equations
edited Mar 27 at 19:04
Maria Mazur
49k1360122
asked Mar 27 at 18:57
ftorftor
113
$endgroup$
$$frac1-fBig(fracxx+(1-x)f(x)Big)1-f(x) = 1-x(1-x)fracf'(x)f(x)$$
Now, we know that $f(x)=c$ and $f(x) = fraca+bx1-x$ are two solutions. How can I get other solutions or to prove that these are all the solutions? Thanks!
functional-equations
functional-equations
edited Mar 27 at 19:04
Maria Mazur
49k1360122
asked Mar 27 at 18:57
ftorftor
113
edited Mar 27 at 19:04
Maria Mazur
49k1360122
asked Mar 27 at 18:57
ftorftor
113
edited Mar 27 at 19:04
Maria Mazur
49k1360122
edited Mar 27 at 19:04
Maria Mazur
49k1360122
edited Mar 27 at 19:04
Maria Mazur
49k1360122
49k1360122
asked Mar 27 at 18:57
ftorftor
113
asked Mar 27 at 18:57
ftorftor
113
asked Mar 27 at 18:57
ftorftor
113
113
$begingroup$
I would substitute $$t=fracxx+(1-x)f(x)$$
$endgroup$
– Dr. Sonnhard Graubner
Mar 27 at 19:02
add a comment |
$begingroup$
I would substitute $$t=fracxx+(1-x)f(x)$$
$endgroup$
– Dr. Sonnhard Graubner
Mar 27 at 19:02
$begingroup$
I would substitute $$t=fracxx+(1-x)f(x)$$
$endgroup$
– Dr. Sonnhard Graubner
Mar 27 at 19:02
$begingroup$
I would substitute $$t=fracxx+(1-x)f(x)$$
$endgroup$
– Dr. Sonnhard Graubner
Mar 27 at 19:02
add a comment |
0
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$begingroup$
I would substitute $$t=fracxx+(1-x)f(x)$$
$endgroup$
– Dr. Sonnhard Graubner
Mar 27 at 19:02