Determination of the series c_i

>   

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

series((-t^6*(c[3]+nu^3))*y^(-6)+O(y^(-5)),y,-5)

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

c3sol := -nu^3

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

series(1/2*t^4*(15*nu^3-2*c[2]-33*nu^2+18*nu)*y^(-4)+O(y^(-3)),y,-3)

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

c2sol := 3/2*nu*(nu-1)*(5*nu-6)

>    factor(subs(K(0)=0,series(subs(c[3]=c3sol,c[2]=c2sol,normal(eqinv)),y,5)));

series(1/2*t^2*(-2*c[1]+12*nu^3-33*nu^2+30*nu-9-9*t^3*nu^2+9*nu^3*t^3)*y^(-2)+O(y^(-1)),y,-1)

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

c1sol := 9/2*nu^2*(nu-1)*t^3+3/2*(4*nu-3)*(nu-1)^2

We ignore the following identity

>    subs(K(0)=0,factor(series(subs(c[3]=c3sol,c[2]=c2sol,c[1]=c1sol,normal(eqinv)),y,6)));

series((-27/2*t*(nu-1)^2*(-nu*t^3+nu*D(K)(0)*t))*y^(-1)+O(1),y,0)

>    subs(K(0)=0,factor(series(subs(c[3]=c3sol,c[2]=c2sol,c[1]=c1sol,normal(eqinv)),y,7)));

series((-27/2*t*(nu-1)^2*(-nu*t^3+nu*D(K)(0)*t))*y^(-1)+(-1+3*nu-3*nu^2+nu^3-9/2*nu^3*t^3+27/2*D(K)(0)*t-9/2*nu*t^3+9*t^3*nu^2-c[0]+27/2*D(K)(0)*t*nu^2-27/4*`@@`(D,2)(K)(0)*t^2*nu-27/4*`@@`(D,2)(K)(0)*...
series((-27/2*t*(nu-1)^2*(-nu*t^3+nu*D(K)(0)*t))*y^(-1)+(-1+3*nu-3*nu^2+nu^3-9/2*nu^3*t^3+27/2*D(K)(0)*t-9/2*nu*t^3+9*t^3*nu^2-c[0]+27/2*D(K)(0)*t*nu^2-27/4*`@@`(D,2)(K)(0)*t^2*nu-27/4*`@@`(D,2)(K)(0)*...

>    c0sol:=collect(solve(coeff(%,y,0),c[0]),[K,t],factor);

c0sol := 27/2*t*(nu-1)^2*D(K)(0)-27/4*nu*(nu-1)^2*t^2*`@@`(D,2)(K)(0)-9/2*nu*(nu-1)^2*t^3+(nu-1)^3

Equation in K(y) with a single catalytic variable

>    catK1:=factor(subs(c[3]=c3sol,c[2]=c2sol,c[1]=c1sol,c[0]=c0sol,eqinv));

catK1 := -27/4*(nu-1)^2*(2*K(y)*t^2*nu-2*y*t^4*nu+2*y^2*K(y)^2*nu-nu*t^2*`@@`(D,2)(K)(0)*y^2+2*t*D(K)(0)*y^2-2*y*K(y)*t)/y^2

Back to the series  Q(0,y)

>    catQ1:=factor(subs(K(y)=t^2*y*Q(y),D(K)(0)=t^2*Q(0),`@@`(D,2)(K)(0)=2*t^2*D(Q)(0),catK1));

catQ1 := -27/2*(nu-1)^2/y*t^3*(t*Q(y)*nu-t*nu+y^3*t*Q(y)^2*nu-nu*t*D(Q)(0)*y+Q(0)*y-y*Q(y))

>    collect(-catQ1/27*2/(nu-1)^2/t^3,Q,factor);

Q(0)+nu*t*y^2*Q(y)^2+1/y*(t*nu-y)*Q(y)-t*nu*D(Q)(0)-1/y*t*nu

>    collect(%+Q(y),Q,factor);

Q(0)+nu*t*y^2*Q(y)^2+1/y*t*nu*Q(y)-t*nu*D(Q)(0)-1/y*t*nu

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