Internal problem ID [10078]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1756 (book 6.165).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime } y a +{y^{\prime }}^{2} b +\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y=-\operatorname {c0}} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 419
dsolve(a*y(x)*diff(diff(y(x),x),x)+b*diff(y(x),x)^2+c4*y(x)^4+c3*y(x)^3+c2*y(x)^2+c1*y(x)+c0=0,y(x), singsol=all)
\begin{align*} 6 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \left (a +\frac {2 b}{3}\right ) \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{\frac {2 b}{a}}}{\sqrt {-36 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \,\textit {\_a}^{\frac {2 b}{a}} \left (\frac {2 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \operatorname {c3} \,\textit {\_a}^{\frac {3 a +2 b}{a}}}{3}+\left (\left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \operatorname {c2} \,\textit {\_a}^{\frac {2 a +2 b}{a}}+\left (a +b \right ) \left (\frac {\operatorname {c4} b \left (a +2 b \right ) \textit {\_a}^{\frac {4 a +2 b}{a}}}{2}+\left (2 \textit {\_a}^{\frac {a +2 b}{a}} b \operatorname {c1} +\left (\textit {\_a}^{\frac {2 b}{a}} \operatorname {c0} -c_{1} b \right ) \left (a +2 b \right )\right ) \left (a +\frac {b}{2}\right )\right )\right ) \left (a +\frac {2 b}{3}\right )\right ) \left (a +2 b \right ) \left (a +\frac {2 b}{3}\right )}}d \textit {\_a} \right )-c_{2} -x &= 0 \\ -6 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \left (a +\frac {2 b}{3}\right ) \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{\frac {2 b}{a}}}{\sqrt {-36 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \,\textit {\_a}^{\frac {2 b}{a}} \left (\frac {2 \left (a +b \right ) \left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \operatorname {c3} \,\textit {\_a}^{\frac {3 a +2 b}{a}}}{3}+\left (\left (a +\frac {b}{2}\right ) b \left (a +2 b \right ) \operatorname {c2} \,\textit {\_a}^{\frac {2 a +2 b}{a}}+\left (a +b \right ) \left (\frac {\operatorname {c4} b \left (a +2 b \right ) \textit {\_a}^{\frac {4 a +2 b}{a}}}{2}+\left (2 \textit {\_a}^{\frac {a +2 b}{a}} b \operatorname {c1} +\left (\textit {\_a}^{\frac {2 b}{a}} \operatorname {c0} -c_{1} b \right ) \left (a +2 b \right )\right ) \left (a +\frac {b}{2}\right )\right )\right ) \left (a +\frac {2 b}{3}\right )\right ) \left (a +2 b \right ) \left (a +\frac {2 b}{3}\right )}}d \textit {\_a} \right )-c_{2} -x &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 12.68 (sec). Leaf size: 2166
DSolve[c0 + c1*y[x] + c2*y[x]^2 + c3*y[x]^3 + c4*y[x]^4 + b*y'[x]^2 + a*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
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