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Daniele Tampieri
Daniele Tampieri
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why riccati Why the Riccati equation \frac$\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementary functional solution "only" when m=0$m=0$,m= $m=-22$,m=4k $m=4k/2k+-1(2k\pm 1)$?

theThe special form of Riccati equation \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} $$ \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} $$ has been proved that it is solvable if and only if m=0,m=-2$m=0$,m=4k/(2k+1) $m=-2$,m=4k/(2k-1) $m=4k/(2k\pm 1)$. 
the sufficencyThe sufficiency is obviously.But But how to prove it'sits necessity?

why riccati equation \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} has an elementary functional solution "only" when m=0,m=-2,m=4k/2k+-1

the special form of Riccati equation \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} has been proved that it is solvable if and only if m=0,m=-2,m=4k/(2k+1),m=4k/(2k-1). the sufficency is obviously.But how to prove it's necessity?

Why the Riccati equation $\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementary solution "only" when $m=0$, $m=-2$, $m=4k/(2k\pm 1)$?

The special form of Riccati equation $$ \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} $$ has been proved that it is solvable if and only if $m=0$, $m=-2$, $m=4k/(2k\pm 1)$. 
The sufficiency is obviously. But how to prove its necessity?

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z yuli
z yuli
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why riccati equation \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} has an elementary functional solution "only" when m=0,m=-2,m=4k/2k+-1

the special form of Riccati equation \frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2} has been proved that it is solvable if and only if m=0,m=-2,m=4k/(2k+1),m=4k/(2k-1). the sufficency is obviously.But how to prove it's necessity?