Internal problem ID [2839]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 4
Problem number: 84.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _Abel]
Solve \begin {gather*} \boxed {y^{\prime }-\left (a +b x y\right ) y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 103
dsolve(diff(y(x),x) = (a+b*x*y(x))*y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\RootOf \left (2 \sqrt {a^{2}-4 b}\, a \arctanh \left (\frac {2 b \,{\mathrm e}^{\textit {\_Z}}+a}{\sqrt {a^{2}-4 b}}\right )-\ln \left (x^{2} \left (b \,{\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) a^{2}+2 c_{1} a^{2}+2 \textit {\_Z} \,a^{2}+4 \ln \left (x^{2} \left (b \,{\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) b -8 c_{1} b -8 \textit {\_Z} b \right )}}{x} \]
✓ Solution by Mathematica
Time used: 0.204 (sec). Leaf size: 94
DSolve[y'[x]==(a+b x y[x])y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {a^2 \left (-\frac {2 \text {ArcTan}\left (\frac {a+2 b x y(x)}{a \sqrt {\frac {4 b}{a^2}-1}}\right )}{\sqrt {\frac {4 b}{a^2}-1}}-\log \left (\frac {b x y(x) (a+b x y(x))+b}{b^2 x^2 y(x)^2}\right )\right )}{2 b}=\frac {a^2 \log (x)}{b}+c_1,y(x)\right ] \]