Determination of the series c_i. An equation with one catalytic variable and 2 additional unknown series Q'(0) and Q'''(0)

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>    factor(series(normal(eqinv),y,1));

series((-t^8*(4*nu^4+c[4]))*y^(-8)+O(y^(-7)),y,-7)

>    c4sol:=factor(solve(coeff(%,y,-8),c[4]));

c4sol := -4*nu^4

>    factor(series(subs(c[4]=c4sol,normal(eqinv)),y,3));

series(t^6*(-c[3]-40*nu^3+16*nu^4+24*nu^2)*y^(-6)+O(y^(-5)),y,-5)

>    c3sol:=factor(solve(coeff(%,y,-6),c[3]));

c3sol := 8*nu^2*(nu-1)*(2*nu-3)

>    factor(series(subs(c[4]=c4sol,c[3]=c3sol,normal(eqinv)),y,5));

series(t^4*(-8*nu+16*t^3*nu^4-c[2]-4+20*nu^4-16*t^3*nu^3-56*nu^3+48*nu^2)*y^(-4)+O(y^(-3)),y,-3)

>    c2sol:=collect(solve(coeff(%,y,-4),c[2]),t,factor);

c2sol := 16*nu^3*(nu-1)*t^3+4*(5*nu+1)*(nu-1)^3

>    factor(series(subs(c[4]=c4sol,c[3]=c3sol,c[2]=c2sol,normal(eqinv)),y,6));

series((-64*t^6*nu^2*(nu-1)^2*(-1+Q(0)))*y^(-3)+O(y^(-2)),y,-2)

>    factor(subs(Q(0)=1,series(subs(c[4]=c4sol,c[3]=c3sol,c[2]=c2sol,normal(eqinv)),y,7)));

series((-t^2*(96*t^3*nu^2-48*t^3*nu^3+c[1]-4-128*t^4*D(Q)(0)*nu^3+64*t^4*D(Q)(0)*nu^4+64*t^4*D(Q)(0)*nu^2-48*t^3*nu-8*nu^4+28*nu^3-36*nu^2+20*nu))*y^(-2)+O(y^(-1)),y,-1)

>    c1sol:=collect(solve(coeff(%,y,-2),c[1]),[Q,t],factor);

c1sol := -64*nu^2*(nu-1)^2*t^4*D(Q)(0)+48*nu*(nu-1)^2*t^3+4*(-1+2*nu)*(nu-1)^3

>    res:=subs(Q(0)=1,factor(series(subs(c[4]=c4sol,c[3]=c3sol,c[2]=c2sol,c[1]=c1sol,normal(eqinv)),y,9)));

res := series((-32*t^4*nu*(nu-1)^2*(nu*t^2*`@@`(D,2)(Q)(0)-2*t*D(Q)(0)))*y^(-1)+(1+6*nu^2-4*nu-8*t^3*nu^4+56*t^3*nu^3-112*t^6*nu^2+32*t^5*`@@`(D,2)(Q)(0)*nu+224*t^6*nu^3-112*t^6*nu^4-4*nu^3-32*t^3+nu^4...
res := series((-32*t^4*nu*(nu-1)^2*(nu*t^2*`@@`(D,2)(Q)(0)-2*t*D(Q)(0)))*y^(-1)+(1+6*nu^2-4*nu-8*t^3*nu^4+56*t^3*nu^3-112*t^6*nu^2+32*t^5*`@@`(D,2)(Q)(0)*nu+224*t^6*nu^3-112*t^6*nu^4-4*nu^3-32*t^3+nu^4...
res := series((-32*t^4*nu*(nu-1)^2*(nu*t^2*`@@`(D,2)(Q)(0)-2*t*D(Q)(0)))*y^(-1)+(1+6*nu^2-4*nu-8*t^3*nu^4+56*t^3*nu^3-112*t^6*nu^2+32*t^5*`@@`(D,2)(Q)(0)*nu+224*t^6*nu^3-112*t^6*nu^4-4*nu^3-32*t^3+nu^4...

>    D2Qsol:=factor(solve(coeff(res,y,-1),`@@`(D,2)(Q)(0)));

D2Qsol := 2*D(Q)(0)/t/nu

>    c0sol:=collect(solve(subs(`@@`(D,2)(Q)(0)=D2Qsol,coeff(res,y,0)),c[0]),[Q,t],factor);

c0sol := -32/3*nu^2*(nu-1)^2*t^6*`@@`(D,3)(Q)(0)-32*(nu+1)*(nu-2)*(nu-1)^2*t^4*D(Q)(0)-112*nu^2*(nu-1)^2*t^6-8*(nu-4)*(nu-1)^3*t^3+(nu-1)^4

>   

The equation with one catalytic variable in Q(y)

>    factor(subs(c[4]=c4sol,c[3]=c3sol,c[2]=c2sol,c[1]=c1sol,c[0]=c0sol,eqinv));

-32/3*t^3*(nu-1)^2*(-3*y^3-12*t^3*y^3*nu^2-6*t^3*y*D(Q)(0)*nu^2+6*t^2*y^2*D(Q)(0)*nu^2+3*y^3*Q(y)-12*t^3*y^4*D(Q)(0)*nu^2*Q(y)-12*t^2*y^4*Q(y)^2*nu-12*t^2*y^4*Q(y)^2*nu^2+24*t^3*y^6*Q(y)^3*nu^2-3*y^3*Q...
-32/3*t^3*(nu-1)^2*(-3*y^3-12*t^3*y^3*nu^2-6*t^3*y*D(Q)(0)*nu^2+6*t^2*y^2*D(Q)(0)*nu^2+3*y^3*Q(y)-12*t^3*y^4*D(Q)(0)*nu^2*Q(y)-12*t^2*y^4*Q(y)^2*nu-12*t^2*y^4*Q(y)^2*nu^2+24*t^3*y^6*Q(y)^3*nu^2-3*y^3*Q...
-32/3*t^3*(nu-1)^2*(-3*y^3-12*t^3*y^3*nu^2-6*t^3*y*D(Q)(0)*nu^2+6*t^2*y^2*D(Q)(0)*nu^2+3*y^3*Q(y)-12*t^3*y^4*D(Q)(0)*nu^2*Q(y)-12*t^2*y^4*Q(y)^2*nu-12*t^2*y^4*Q(y)^2*nu^2+24*t^3*y^6*Q(y)^3*nu^2-3*y^3*Q...
-32/3*t^3*(nu-1)^2*(-3*y^3-12*t^3*y^3*nu^2-6*t^3*y*D(Q)(0)*nu^2+6*t^2*y^2*D(Q)(0)*nu^2+3*y^3*Q(y)-12*t^3*y^4*D(Q)(0)*nu^2*Q(y)-12*t^2*y^4*Q(y)^2*nu-12*t^2*y^4*Q(y)^2*nu^2+24*t^3*y^6*Q(y)^3*nu^2-3*y^3*Q...

>    op(4,%);

-3*y^3-12*t^3*y^3*nu^2-6*t^3*y*D(Q)(0)*nu^2+6*t^2*y^2*D(Q)(0)*nu^2+3*y^3*Q(y)-12*t^3*y^4*D(Q)(0)*nu^2*Q(y)-12*t^2*y^4*Q(y)^2*nu-12*t^2*y^4*Q(y)^2*nu^2+24*t^3*y^6*Q(y)^3*nu^2-3*y^3*Q(y)*nu+6*t*D(Q)(0)*y...
-3*y^3-12*t^3*y^3*nu^2-6*t^3*y*D(Q)(0)*nu^2+6*t^2*y^2*D(Q)(0)*nu^2+3*y^3*Q(y)-12*t^3*y^4*D(Q)(0)*nu^2*Q(y)-12*t^2*y^4*Q(y)^2*nu-12*t^2*y^4*Q(y)^2*nu^2+24*t^3*y^6*Q(y)^3*nu^2-3*y^3*Q(y)*nu+6*t*D(Q)(0)*y...
-3*y^3-12*t^3*y^3*nu^2-6*t^3*y*D(Q)(0)*nu^2+6*t^2*y^2*D(Q)(0)*nu^2+3*y^3*Q(y)-12*t^3*y^4*D(Q)(0)*nu^2*Q(y)-12*t^2*y^4*Q(y)^2*nu-12*t^2*y^4*Q(y)^2*nu^2+24*t^3*y^6*Q(y)^3*nu^2-3*y^3*Q(y)*nu+6*t*D(Q)(0)*y...
-3*y^3-12*t^3*y^3*nu^2-6*t^3*y*D(Q)(0)*nu^2+6*t^2*y^2*D(Q)(0)*nu^2+3*y^3*Q(y)-12*t^3*y^4*D(Q)(0)*nu^2*Q(y)-12*t^2*y^4*Q(y)^2*nu-12*t^2*y^4*Q(y)^2*nu^2+24*t^3*y^6*Q(y)^3*nu^2-3*y^3*Q(y)*nu+6*t*D(Q)(0)*y...

>    indets(%);

{y, t, nu, Q(y)}

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