> restart;
> with(DEtools);

  [AreSimilar, DEnormal, DEplot, DEplot3d, DEplot_polygon, DFactor,

        DFactorLCLM, DFactorsols, Dchangevar, FunctionDecomposition,

        GCRD, Gosper, Heunsols, Homomorphisms, IsHyperexponential,

        LCLM, MeijerGsols, MultiplicativeDecomposition, ODEInvariants,

        PDEchangecoords, PolynomialNormalForm, RationalCanonicalForm,

        ReduceHyperexp, RiemannPsols, Xchange, Xcommutator, Xgauge,

        Zeilberger, abelsol, adjoint, autonomous, bernoullisol,

        buildsol, buildsym, canoni, caseplot, casesplit, checkrank,

        chinisol, clairautsol, constcoeffsols, convertAlg, convertsys,

        dalembertsol, dcoeffs, de2diffop, dfieldplot, diff_table,

        diffop2de, dperiodic_sols, dpolyform, dsubs, eigenring,

        endomorphism_charpoly, equinv, eta_k, eulersols, exactsol,

        expsols, exterior_power, firint, firtest, formal_sol, gen_exp,

        generate_ic, genhomosol, gensys, hamilton_eqs, hypergeomsols,

        hyperode, indicialeq, infgen, initialdata, integrate_sols,

        intfactor, invariants, kovacicsols, leftdivision, liesol,

        line_int, linearsol, matrixDE, matrix_riccati, maxdimsystems,

        moser_reduce, muchange, mult, mutest, newton_polygon,

        normalG2, ode_int_y, ode_y1, odeadvisor, odepde,

        parametricsol, particularsol, phaseportrait, poincare,

        polysols, power_equivalent, rational_equivalent, ratsols,

        redode, reduceOrder, reduce_order, regular_parts, regularsp,

        remove_RootOf, riccati_system, riccatisol, rifread, rifsimp,

        rightdivision, rtaylor, separablesol, singularities,

        solve_group, super_reduce, symgen, symmetric_power,

        symmetric_product, symtest, transinv, translate, untranslate,

        varparam, zoom]

> sys:=diff(x(t),t)=(2*(x(t))^2+3*x(t)+1)/y(t),diff(y(t),t)=-2*x(t)-2;

                            2
            d         2 x(t)  + 3 x(t) + 1  d
     sys := -- x(t) = --------------------, -- y(t) = -2 x(t) - 2
            dt                y(t)          dt

> ic:=x(1)=-2/3,y(1)=-3/2;

                    ic := x(1) = -2/3, y(1) = -3/2

> dsolve([sys,ic]);
> DEplot( [sys], [x(t),y(t)], t=-60..40, x=-1..-0.5, y=-35..1,
>   [[ic]], arrows=medium);

> ed:=diff(y(t),t)=-1/y(t)/2-1;

                          d               1
                    ed := -- y(t) = -1/2 ---- - 1
                          dt             y(t)

> ic1:=y(1)=-3/2;

                          ic1 := y(1) = -3/2

> yy:=subs(dsolve([ed,ic1],y(t)),y(t));

         yy := - 1/2 - 1/2 LambertW(-2 exp(-I (2 I t - Pi)))

> xx:=-1/2+1/4/yy;

                                           1
   xx := - 1/2 + 1/4 ---------------------------------------------
                     - 1/2 - 1/2 LambertW(-2 exp(-I (2 I t - Pi)))

> plot([xx,yy,t=-60..40]);

> series(yy,t=0);

                                LambertW(2)          LambertW(2)
  (- 1/2 - 1/2 LambertW(2)) - --------------- t - ------------------
                              1 + LambertW(2)                      3
                                                  (1 + LambertW(2))

         2       LambertW(2) (2 LambertW(2) - 1)  3
        t  + 2/3 ------------------------------- t  -
                                        5
                       (1 + LambertW(2))

                                                       2
            LambertW(2) (-8 LambertW(2) + 6 LambertW(2)  + 1)  4
        1/3 ------------------------------------------------- t  +
                                            7
                           (1 + LambertW(2))

        2/15 LambertW(2)

                                            2                 3    /
        (22 LambertW(2) - 1 - 58 LambertW(2)  + 24 LambertW(2) )  /
                                                                 /

                         9  5      6
        (1 + LambertW(2))  t  + O(t )

>