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ode:=diff(y(x),x$2)=(diff(y(x),x))^3-(diff(y(x),x))^2;
IC:=y(0)=3,D(y)(0)=1;
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Error, (in dsolve) numeric exception: division by zero
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sol:=y(x)=3+x;
odetest(sol,[ode,IC])
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![[0, 0, 0]](/view.aspx?sf=238756_question/9712e3955d41be5ec2c5ed05d2394480.gif)
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DEtools:-odeadvisor(ode);
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![[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]](/view.aspx?sf=238756_question/cdeec7f53d6b3100296d4fec951c1e31.gif)

Methods for second order ODEs:
--- Trying classification methods ---
trying 2nd order Liouville
trying 2nd order WeierstrassP
trying 2nd order JacobiSN
differential order: 2; trying a linearization to 3rd order
--- trying a change of variables {x -> y(x), y(x) -> x}
differential order: 2; trying a linearization to 3rd order
trying 2nd order ODE linearizable_by_differentiation
trying 2nd order, 2 integrating factors of the form mu(x,y)
trying differential order: 2; missing variables
-> Computing canonical coordinates for the symmetry [0, 1]
-> Rewriting ODE in canonical coordinates by means of differential invariants
Try computing 1 more symmetries for ODE written in canonical coordinates
-> Computing symmetries using: way = 3
Found another symmetry:
![[1, 0]](/view.aspx?sf=238756_question/55bbc5ade51d448edc802a03580fb498.gif)
Found another symmetry:
![[-x+y, 0]](/view.aspx?sf=238756_question/6a62767825cd3901f6e98f3c5fab6161.gif)
Computing a convenient ordering to use the 3 symmetries available
<- differential order: 2; canonical coordinates successful
<- differential order 2; missing variables successful
Error, (in dsolve) numeric exception: division by zero
Methods for second order ODEs:
--- Trying classification methods ---
trying 2nd order Liouville
trying 2nd order WeierstrassP
trying 2nd order JacobiSN
differential order: 2; trying a linearization to 3rd order
--- trying a change of variables {x -> y(x), y(x) -> x}
differential order: 2; trying a linearization to 3rd order
trying 2nd order ODE linearizable_by_differentiation
trying 2nd order, 2 integrating factors of the form mu(x,y)
trying differential order: 2; missing variables
-> Computing canonical coordinates for the symmetry [0, 1]
-> Rewriting ODE in canonical coordinates by means of differential invariants
Try computing 1 more symmetries for ODE written in canonical coordinates
-> Computing symmetries using: way = 3
Found another symmetry:
![[1, 0]](/view.aspx?sf=238756_question/ad30890ddcc8cb817424793ae0148a4e.gif)
Found another symmetry:
![[-x+y, 0]](/view.aspx?sf=238756_question/2159b09b92b3810f131b742dfd846521.gif)
Computing a convenient ordering to use the 3 symmetries available
-> Calling odsolve with the ODE diff(_b(_a) _a) = _b(_a)^3-_b(_a)^2 _b(_a) HINT = [[1 0] [-_a+y -_b*(_b-1)]]
*** Sublevel 2 ***
symmetry methods on request
1st order, trying reduction of order with given symmetries:
![[1, 0], [-_a+y, -_b*(_b-1)]](/view.aspx?sf=238756_question/b643180896365966aec6f692fd89ade2.gif)
1st order, trying the canonical coordinates of the invariance group
<- 1st order, canonical coordinates successful
<- differential order: 2; canonical coordinates successful
<- differential order 2; missing variables successful

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