As the first example we consider here the one presented in Kamke's book with number 185
If infolevel is set to a greater integer (possible settings are 1 through 5), more detailed information about the computation method is displayed.
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Methods for first order ODEs:
 Trying classification methods 
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The relative invariant s3 is: 10/27/x^12*(9*x^2+2)
The first absolute invariant s5^3/s3^5 is: 729/100*(90*x^415*x^214)^3/(9*x^2+2)^5
The second absolute invariant s3*s7/s5^2 is: 5/3*(9*x^2+2)*(972*x^6324*x^415*x^2+98)/(90*x^415*x^214)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
inverse of the transformation solving the problem is: {t = x, u(t) = y(x)}
< Abel successful
 
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The integrals above can be evaluated in terms of hypergeometric functions using value. These implicit results can be tested using odetest
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The above means mainly that dsolve can solve the whole class associated to this ODE. For instance, by changing variables
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and this ODE is also solvable (actually for arbitrary F(t), P(t) and Q(t)):
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Methods for first order ODEs:
 Trying classification methods 
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The first invariant is nonrational: 729/100*(90*F(x)^415*F(x)^214)^3/(9*F(x)^2+2)^5
> Searching for a convenient change of variables...
< Unable to rationalize the invariant
The relative invariant s3 is: 10/27*(9*F(x)^2+2)/F(x)^12*diff(F(x),x)^3*P(x)^3
The first absolute invariant s5^3/s3^5 is: 729/100*(90*F(x)^415*F(x)^214)^3/(9*F(x)^2+2)^5
The second absolute invariant s3*s7/s5^2 is: 5/3*(9*F(x)^2+2)*(972*F(x)^6324*F(x)^415*F(x)^2+98)/(90*F(x)^415*F(x)^214)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
inverse of the transformation solving the problem is: {t = F(x), u(t) = P(x)*y(x)+Q(x)}
< Abel successful
 
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The next example is still from Kamke's and appears there with number 257
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Methods for first order ODEs:
 Trying classification methods 
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
trying Abel
The equivalent Abel ODE of 1st kind is: diff(u(x),x) = 2*x*(x^2+1)*(x^21)*u(x)^32/x^2*u(x)^21/x*u(x)
The relative invariant s3 is: 8/27*(27*x^89*x^4+2)/x^6
The first absolute invariant s5^3/s3^5 is: 729*(135*x^1636*x^12+54*x^815*x^4+2)^3/(27*x^89*x^4+2)^5
The second absolute invariant s3*s7/s5^2 is: 1/3*(27*x^89*x^4+2)*(2835*x^24+243*x^20+2349*x^16927*x^12+495*x^8105*x^4+10)/(135*x^1636*x^12+54*x^815*x^4+2)^2
...checking Abel class AIL (45)
...checking Abel class AIL (310)
...checking Abel class AIR (36)
...checking Abel class AIL (301)
...checking Abel class AIL (1000)
...checking Abel class AIL (42)
...checking Abel class AIL (185)
...checking Abel class AIA (by Halphen)
...checking Abel class AIL (205)
...checking Abel class AIA (147)
...checking Abel class AIL (581)
...checking Abel class AIL (200)
...checking Abel class AIL (257)
inverse of the transformation solving the problem is: {t = 1/x^2, u(t) = x*y(x)}
< Abel successful
 
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This ODE actually belongs to the Abel class represented by the simpler ODE
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Actually, by converting ode[257] from Second Kind to Abel First Kind format
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then changing variables in this ODE above (first element in the sequence) using
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and renaming the variables x=t, u=y we obtain:
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Isolating y' we arrive at the ODE representative for the Abel Class[257]
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A much more complicated example is given by the ODE presented in Kamke's book with number 43:
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This ODE belongs to class B (by Liouville); that is: it can be obtained from
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by changing variables
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and setting the parameter C in the Abel ODE representative of Class[B] as
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The result above is in fact ode[43]. The process of determining the value of C for which a equivalence between the ODEs Class[B] and ode[43] exists, as well as the explicit form of the equivalence transformation, followed by using it to build the answer to ode[43] is now available via
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The database of solvable classes dsolve includes representatives for 25 classes. To each class there is associated a number. These numbers can be seen via
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Classes 1.1 to 1.9 are parameterized classes (parameter C), and each class representative  for instance, for Class 33  can be seen via
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