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differential equations - Should DSolve always return solution with constant of integration?


Bug introduced in 10.0.0 and fixed in 10.0.2



Clear[y,x];
DSolve[D[y[x], x] - y[x]^2 + y[x]*Sin[x] - Cos[x] == 0, y[x], x, 

    GeneratedParameters -> C]

or


DSolve[D[y[x], x] - y[x]^2 + y[x]*Sin[x] - Cos[x] == 0, y[x], x]

both return a solution that does not include C[1], the constant of integration.


{{y[x] -> Sin[x]}}

The question is: Should DSolve always return an arbitrary constant? Even though the answer is correct, it is missing C[1] hence this is a particular solution.


If DSolve does not have to generate a constant of integration in the solution of a differential equation, then what caused it not to generate it in this specific case?



Update:


Let me add a solution found by Maple for this, which does include a constant of integration:


Clear[C,y,x];
eq = Derivative[1][y][x] - y[x]^2 + y[x]*Sin[x] - Cos[x] == 0;
eq /. y -> (- Exp[-Cos[#]]/(C[1] + Integrate[ Exp[-Cos[#]], x]) + Sin[#] &);
Simplify[%]
(* True *)

So, the above is a general solution with a constant of integration that solves the same differential equation.



Answer




I think it's a bug.


Tracing


Tracing the evaluation of DSolve as the following:


eq = D[y[x], x] - y[x]^2 + y[x]*Sin[x] - Cos[x] == 0;

traceRes = Trace[DSolve[eq, y[x], x, 
                        GeneratedParameters -> ThisIsForGeneralC],
                 {TraceInternal -> True,
                 TraceOff -> _Message}];


and formatting (using the levelIndentFunc function I mentioned here) and exporting the result (it will be around 200 MByte):


Export["[Trace-result] DSolve.txt", levelIndentFunc @ traceRes, "String"]

Searching the "unique" footprint ThisIsForGeneralC we made on purpose, it's not hard to find where the problem comes from.


Analysis


Here, from the trace result we can see, MMA eventually arrives a point like:


DSolve`DSolveFirstOrderODEDump`f = {{{y[x] -> E^Cos[x]*DSolve`DSolveFirstOrderODEDump`const[2] + E^Cos[x]*Integrate[DSolve`DSolveFirstOrderODEDump`const[1]/E^Cos[K[1]], {K[1], 1, x}]}}}
DSolve`DSolveFirstOrderODEDump`f = DSolve`DSolveFirstOrderODEDump`f[[1,1,1,2]]; {DSolve`DSolveFirstOrderODEDump`f, DSolve`DSolveFirstOrderODEDump`g} = {Coefficient[DSolve`DSolveFirstOrderODEDump`f, DSolve`DSolveFirstOrderODEDump`const[1]], Coefficient[DSolve`DSolveFirstOrderODEDump`f, DSolve`DSolveFirstOrderODEDump`const[2]]}; {{y[x] -> -((ThisIsForGeneralC[1]*D[DSolve`DSolveFirstOrderODEDump`f, x] + D[DSolve`DSolveFirstOrderODEDump`g, x])/(DSolve`DSolveFirstOrderODEDump`h*(ThisIsForGeneralC[1]*DSolve`DSolveFirstOrderODEDump`f + DSolve`DSolveFirstOrderODEDump`g)))}}

which is actually



f = {{{y[x] -> E^Cos[x]*const[2] + E^Cos[x]*Integrate[const[1]/E^Cos[K[1]], {K[1], 1, x}]}}}; 
f = f[[1, 1, 1, 2]]

intermedia f


{f, g} = {Coefficient[f, const[1]], Coefficient[f, const[2]]}
{{y[x] -> -((ThisIsForGeneralC[1]*D[f, x] + D[g, x])/(h*(ThisIsForGeneralC[1]*f + g)))}}

final false result


Note the f = Coefficient[f, const[1]] part, which is incorrectly evaluated to 0! That's the one to blame for our issue!


If we replace f with the correct value:



f = E^Cos[x]*Integrate[E^(-Cos[K[1]]), {K[1], 1, x}];

We'll get effectively the same general solution as the one mentioned in OP:


{{y[x] -> -((ThisIsForGeneralC[1]*D[f, x] + D[g, x])/(h*(ThisIsForGeneralC[1]*f + g)))}}

correct result


Some perhaps fixes coming up to my mind include:



  1. Introducing new rule for Integrate (Please compare Integrate[a b[x], {x, 0, 1}] and Integrate[a b[x], x]); Or

  2. Introducing new rule for Coefficient (Maybe not a good idea); Or


  3. Using method other than Coefficient in DSolve.


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