Internal problem ID [10126]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1804 (book 6.213).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {2 \left (y-a \right ) \left (y-b \right ) \left (y-c \right ) y^{\prime \prime }-\left (\left (y-a \right )^{2} \left (y-b \right ) \left (y-c \right )+\left (y-b \right ) \left (y-c \right )\right ) {y^{\prime }}^{2}+\left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2} \left (A_{0} +\frac {B_{0}}{\left (y-a \right )^{2}}+\frac {C_{1}}{\left (y-b \right )^{2}}+\frac {D_{0}}{\left (y-c \right )^{2}}\right )=0} \]
✓ Solution by Maple
Time used: 40.172 (sec). Leaf size: 1097
dsolve(2*(y(x)-a)*(y(x)-b)*(y(x)-c)*diff(y(x),x$2)-( (y(x)-a)*(y(x)-b)*(y(x)-a)*(y(x)-c)+(y(x)-b)*(y(x)-c) )*diff(y(x),x)^2+( (y(x)-a)*(y(x)-b)*(y(x)-c) )^2*(A__0+B__0/(y(x)-a)^2+C__1/(y(x)-b)^2+D__0/(y(x)-c)^2)=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\sqrt {\left (a -\textit {\_m} \right ) \left (\left (\left (\left (a^{2}+4 a b +b^{2}\right ) A_{0} +B_{0} +C_{1} \right ) c^{2}+\left (4 a b \left (a +b \right ) A_{0} +4 C_{1} a +4 B_{0} b \right ) c +A_{0} a^{2} b^{2}+\left (B_{0} +D_{0} \right ) b^{2}+4 D_{0} a b +a^{2} \left (C_{1} +D_{0} \right )\right ) \left (\int \frac {\textit {\_m}^{2} {\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+\left (-2 A_{0} \left (a +b \right ) c^{2}+\left (\left (-2 a^{2}-8 a b -2 b^{2}\right ) A_{0} -2 B_{0} -2 C_{1} \right ) c -2 a b \left (a +b \right ) A_{0} +\left (-2 B_{0} -2 D_{0} \right ) b -2 a \left (C_{1} +D_{0} \right )\right ) \left (\int \frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}} \textit {\_m}^{3}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+\left (c^{2} A_{0} +4 A_{0} \left (a +b \right ) c +\left (a^{2}+4 a b +b^{2}\right ) A_{0} +B_{0} +C_{1} +D_{0} \right ) \left (\int \frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}} \textit {\_m}^{4}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )-2 A_{0} \left (a +b +c \right ) \left (\int \frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}} \textit {\_m}^{5}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+A_{0} \left (\int \frac {\textit {\_m}^{6} {\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+\left (\left (-2 a b \left (a +b \right ) A_{0} -2 C_{1} a -2 B_{0} b \right ) c^{2}+\left (-2 A_{0} a^{2} b^{2}-2 B_{0} b^{2}-2 C_{1} a^{2}\right ) c -2 b D_{0} a \left (a +b \right )\right ) \left (\int \frac {\textit {\_m} \,{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+\left (\left (A_{0} a^{2} b^{2}+B_{0} b^{2}+C_{1} a^{2}\right ) c^{2}+D_{0} a^{2} b^{2}\right ) \left (\int \frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )-c_{1} \right ) {\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}}d \textit {\_m} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\sqrt {\left (a -\textit {\_m} \right ) \left (\left (\left (\left (a^{2}+4 a b +b^{2}\right ) A_{0} +B_{0} +C_{1} \right ) c^{2}+\left (4 a b \left (a +b \right ) A_{0} +4 C_{1} a +4 B_{0} b \right ) c +A_{0} a^{2} b^{2}+\left (B_{0} +D_{0} \right ) b^{2}+4 D_{0} a b +a^{2} \left (C_{1} +D_{0} \right )\right ) \left (\int \frac {\textit {\_m}^{2} {\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+\left (-2 A_{0} \left (a +b \right ) c^{2}+\left (\left (-2 a^{2}-8 a b -2 b^{2}\right ) A_{0} -2 B_{0} -2 C_{1} \right ) c -2 a b \left (a +b \right ) A_{0} +\left (-2 B_{0} -2 D_{0} \right ) b -2 a \left (C_{1} +D_{0} \right )\right ) \left (\int \frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}} \textit {\_m}^{3}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+\left (c^{2} A_{0} +4 A_{0} \left (a +b \right ) c +\left (a^{2}+4 a b +b^{2}\right ) A_{0} +B_{0} +C_{1} +D_{0} \right ) \left (\int \frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}} \textit {\_m}^{4}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )-2 A_{0} \left (a +b +c \right ) \left (\int \frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}} \textit {\_m}^{5}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+A_{0} \left (\int \frac {\textit {\_m}^{6} {\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+\left (\left (-2 a b \left (a +b \right ) A_{0} -2 C_{1} a -2 B_{0} b \right ) c^{2}+\left (-2 A_{0} a^{2} b^{2}-2 B_{0} b^{2}-2 C_{1} a^{2}\right ) c -2 b D_{0} a \left (a +b \right )\right ) \left (\int \frac {\textit {\_m} \,{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )+\left (\left (A_{0} a^{2} b^{2}+B_{0} b^{2}+C_{1} a^{2}\right ) c^{2}+D_{0} a^{2} b^{2}\right ) \left (\int \frac {{\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}{\left (a -\textit {\_m} \right )^{2} \left (-\textit {\_m} +c \right ) \left (b -\textit {\_m} \right )}d \textit {\_m} \right )-c_{1} \right ) {\mathrm e}^{-\frac {\textit {\_m} \left (-2 a +\textit {\_m} \right )}{2}}}}d \textit {\_m} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 72.642 (sec). Leaf size: 3800
DSolve[2*(y[x]-a)*(y[x]-b)*(y[x]-c)*y''[x]-( (y[x]-a)*(y[x]-b)*(y[x]-a)*(y[x]-c)+(y[x]-b)*(y[x]-c) )*y'[x]^2+( (y[x]-a)*(y[x]-b)*(y[x]-c) )^2*(A0+B0/(y[x]-a)^2+C1/(y[x]-b)^2+D0/(y[x]-c)^2)==0,y[x],x,IncludeSingularSolutions -> True]
Too large to display