Internal problem ID [9285]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 950.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Abel]
\[ \boxed {y^{\prime }-y^{2}-\frac {a \,x^{2} y}{2}-y b x -y^{3}-\frac {3 y^{2} a \,x^{2}}{4}-\frac {3 y^{2} b x}{2}-\frac {3 y a^{2} x^{4}}{16}-\frac {3 y a \,x^{3} b}{4}-\frac {3 y b^{2} x^{2}}{4}=-\frac {1}{2} a x +1+\frac {1}{16} a^{2} x^{4}+\frac {1}{4} a b \,x^{3}+\frac {1}{4} b^{2} x^{2}+\frac {1}{64} a^{3} x^{6}+\frac {3}{32} a^{2} x^{5} b +\frac {3}{16} a \,x^{4} b^{2}+\frac {1}{8} b^{3} x^{3}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 42
dsolve(diff(y(x),x) = -1/2*a*x+1+y(x)^2+1/2*a*x^2*y(x)+b*x*y(x)+1/16*a^2*x^4+1/4*a*x^3*b+1/4*b^2*x^2+y(x)^3+3/4*x^2*a*y(x)^2+3/2*y(x)^2*b*x+3/16*y(x)*a^2*x^4+3/4*y(x)*a*x^3*b+3/4*y(x)*b^2*x^2+1/64*a^3*x^6+3/32*a^2*x^5*b+3/16*a*x^4*b^2+1/8*b^3*x^3,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {a \,x^{2}}{4}-\frac {x b}{2}+\operatorname {RootOf}\left (-x +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+b +2}d \textit {\_a} \right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 6.716 (sec). Leaf size: 920
DSolve[y'[x] == 1 - (a*x)/2 + (b^2*x^2)/4 + (a*b*x^3)/4 + (b^3*x^3)/8 + (a^2*x^4)/16 + (3*a*b^2*x^4)/16 + (3*a^2*b*x^5)/32 + (a^3*x^6)/64 + b*x*y[x] + (a*x^2*y[x])/2 + (3*b^2*x^2*y[x])/4 + (3*a*b*x^3*y[x])/4 + (3*a^2*x^4*y[x])/16 + y[x]^2 + (3*b*x*y[x]^2)/2 + (3*a*x^2*y[x]^2)/4 + y[x]^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{9} \text {RootSum}\left [729 b^2 \text {$\#$1}^9+3132 b \text {$\#$1}^9+3364 \text {$\#$1}^9+2187 b^2 \text {$\#$1}^6+9396 b \text {$\#$1}^6+10092 \text {$\#$1}^6+2187 b^2 \text {$\#$1}^3+9396 b \text {$\#$1}^3+9984 \text {$\#$1}^3+729 b^2+3132 b+3364\&,\frac {729 b^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+3132 b \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+3364 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+81\ 2^{2/3} b \sqrt [3]{27 b+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^4+174\ 2^{2/3} \sqrt [3]{27 b+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^4+1458 b^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+6264 b \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+6728 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+18 \sqrt [3]{2} (27 b+58)^{2/3} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}^2+81\ 2^{2/3} b \sqrt [3]{27 b+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}+174\ 2^{2/3} \sqrt [3]{27 b+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right ) \text {$\#$1}+729 b^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right )+3132 b \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right )+3364 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 a x^2+6 b x+4\right )+3 y(x)\right )}{\sqrt [3]{27 b+58}}-\text {$\#$1}\right )}{729 b^2 \text {$\#$1}^8+3132 b \text {$\#$1}^8+3364 \text {$\#$1}^8+1458 b^2 \text {$\#$1}^5+6264 b \text {$\#$1}^5+6728 \text {$\#$1}^5+729 b^2 \text {$\#$1}^2+3132 b \text {$\#$1}^2+3328 \text {$\#$1}^2}\&\right ]=\frac {(27 b+58)^{2/3} x}{9\ 2^{2/3}}+c_1,y(x)\right ] \]