Internal problem ID [8531]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 951.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Abel]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {x}{2}-1-y^{2}-\frac {y x^{2}}{2}-a y x -\frac {x^{4}}{16}-\frac {x^{3} a}{4}-\frac {a^{2} x^{2}}{4}-y^{3}-\frac {3 y^{2} x^{2}}{4}-\frac {3 a x y^{2}}{2}-\frac {3 y x^{4}}{16}-\frac {3 y a \,x^{3}}{4}-\frac {3 y a^{2} x^{2}}{4}-\frac {x^{6}}{64}-\frac {3 a \,x^{5}}{32}-\frac {3 a^{2} x^{4}}{16}-\frac {a^{3} x^{3}}{8}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 41
dsolve(diff(y(x),x) = -1/2*x+1+y(x)^2+1/2*x^2*y(x)+y(x)*a*x+1/16*x^4+1/4*x^3*a+1/4*a^2*x^2+y(x)^3+3/4*x^2*y(x)^2+3/2*a*x*y(x)^2+3/16*y(x)*x^4+3/4*y(x)*a*x^3+3/4*a^2*x^2*y(x)+1/64*x^6+3/32*x^5*a+3/16*a^2*x^4+1/8*a^3*x^3,y(x), singsol=all)
\[ y \relax (x ) = -\frac {x^{2}}{4}-\frac {a x}{2}+\RootOf \left (-x +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+a +2}d \textit {\_a} \right )+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.261 (sec). Leaf size: 140
DSolve[y'[x] == 1 - x/2 + (a^2*x^2)/4 + (a*x^3)/4 + (a^3*x^3)/8 + x^4/16 + (3*a^2*x^4)/16 + (3*a*x^5)/32 + x^6/64 + a*x*y[x] + (x^2*y[x])/2 + (3*a^2*x^2*y[x])/4 + (3*a*x^3*y[x])/4 + (3*x^4*y[x])/16 + y[x]^2 + (3*a*x*y[x]^2)/2 + (3*x^2*y[x]^2)/4 + y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {1}{3} (27 a+58)^{2/3} \text {RootSum}\left [\text {$\#$1}^3 (27 a+58)^{2/3}-3\ 2^{2/3} \text {$\#$1}+(27 a+58)^{2/3}\&,\frac {\log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (6 a x+3 x^2+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right )}{2^{2/3}-\text {$\#$1}^2 (27 a+58)^{2/3}}\&\right ]=\frac {(27 a+58)^{2/3} x}{9\ 2^{2/3}}+c_1,y(x)\right ] \]