Internal problem ID [9996]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2. Equations of
the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-\left (3 x a +b \right ) y+a^{2} x^{3}+a \,x^{2} b -x c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 937
dsolve(y(x)*diff(y(x),x)=(3*a*x+b)*y(x)-a^2*x^3-a*b*x^2+c*x,y(x), singsol=all)
\[ y \relax (x ) = -\frac {9 x \left (a^{2} b^{2} x^{2}+3 a^{2} c \,x^{2}+a \,b^{3} x +3 a b c x -b^{2} c -3 c^{2}\right )}{-9 a \,b^{2} x -27 c x a +b \,{\mathrm e}^{\RootOf \left (2 a \,b^{8} \arctanh \left (\frac {b^{2} \left (9 b^{2}+27 c +2 \,{\mathrm e}^{\textit {\_Z}}\right )}{9 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}}\right )+20 a \,b^{6} c \arctanh \left (\frac {b^{2} \left (9 b^{2}+27 c +2 \,{\mathrm e}^{\textit {\_Z}}\right )}{9 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}}\right )+66 a \,b^{4} c^{2} \arctanh \left (\frac {b^{2} \left (9 b^{2}+27 c +2 \,{\mathrm e}^{\textit {\_Z}}\right )}{9 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}}\right )+72 a \,b^{2} c^{3} \arctanh \left (\frac {b^{2} \left (9 b^{2}+27 c +2 \,{\mathrm e}^{\textit {\_Z}}\right )}{9 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}}\right )+\sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, \ln \left (\frac {\left (2 b^{2}+9 c \right ) \left (a^{2} x^{2}+a b x -c \right )}{x^{2} \left (9 b^{4} {\mathrm e}^{\textit {\_Z}}-81 b^{4} c +b^{2} {\mathrm e}^{2 \textit {\_Z}}+27 b^{2} c \,{\mathrm e}^{\textit {\_Z}}-486 b^{2} c^{2}-729 c^{3}\right )}\right ) a \,b^{4}+2 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, \textit {\_Z} a \,b^{4}+7 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, \ln \left (\frac {\left (2 b^{2}+9 c \right ) \left (a^{2} x^{2}+a b x -c \right )}{x^{2} \left (9 b^{4} {\mathrm e}^{\textit {\_Z}}-81 b^{4} c +b^{2} {\mathrm e}^{2 \textit {\_Z}}+27 b^{2} c \,{\mathrm e}^{\textit {\_Z}}-486 b^{2} c^{2}-729 c^{3}\right )}\right ) a \,b^{2} c -2 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, \sqrt {b^{2} a^{2}+4 c \,a^{2}}\, b^{3} \arctanh \left (\frac {a \left (2 a x +b \right )}{\sqrt {b^{2} a^{2}+4 c \,a^{2}}}\right )+6 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, c_{1} a \,b^{2} c +14 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, \textit {\_Z} a \,b^{2} c +12 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, \ln \left (\frac {\left (2 b^{2}+9 c \right ) \left (a^{2} x^{2}+a b x -c \right )}{x^{2} \left (9 b^{4} {\mathrm e}^{\textit {\_Z}}-81 b^{4} c +b^{2} {\mathrm e}^{2 \textit {\_Z}}+27 b^{2} c \,{\mathrm e}^{\textit {\_Z}}-486 b^{2} c^{2}-729 c^{3}\right )}\right ) a \,c^{2}-6 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, \sqrt {b^{2} a^{2}+4 c \,a^{2}}\, b c \arctanh \left (\frac {a \left (2 a x +b \right )}{\sqrt {b^{2} a^{2}+4 c \,a^{2}}}\right )+24 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, c_{1} a \,c^{2}+24 \sqrt {b^{8}+10 b^{6} c +33 b^{4} c^{2}+36 b^{2} c^{3}}\, \textit {\_Z} a \,c^{2}\right )}} \]
✓ Solution by Mathematica
Time used: 1.161 (sec). Leaf size: 194
DSolve[y[x]*y'[x]==(3*a*x+b)*y[x]-a^2*x^3-a*b*x^2+c*x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2 a b \left (\text {RootSum}\left [\text {$\#$1}^4 a^2+\text {$\#$1}^3 a b-2 \text {$\#$1}^2 a y(x)-\text {$\#$1}^2 c-\text {$\#$1} b y(x)+y(x)^2\&,\frac {-2 \text {$\#$1}^3 a^2 \log (x-\text {$\#$1})-\text {$\#$1}^2 a b \log (x-\text {$\#$1})+2 \text {$\#$1} a y(x) \log (x-\text {$\#$1})+b y(x) \log (x-\text {$\#$1})+\text {$\#$1} c \log (x-\text {$\#$1})}{-4 \text {$\#$1}^3 a^2-3 \text {$\#$1}^2 a b+4 \text {$\#$1} a y(x)+2 \text {$\#$1} c+b y(x)}\&\right ]-\log \left (a \left (-a x^2-b x+y(x)\right )+c\right )\right )}{c (3 a+b+c-1)}=c_1,y(x)\right ] \]