Internal problem ID [10087]
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]]
\[ \boxed {y^{\prime \prime } y a +b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y=-\operatorname {c0}} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 1028
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*} \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 b \,a^{3}+35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right )}{\sqrt {-\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 b \,a^{3}+35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right ) \left (25 \textit {\_a}^{\frac {2 b}{a}} a^{3} b \operatorname {c0} -35 c_{1} a^{2} b^{3}+4 \operatorname {c3} \,b^{4} \textit {\_a}^{\frac {3 a +2 b}{a}}+12 b \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{3}+26 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{2} b^{2}+18 a \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{3}+4 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{4}+6 \textit {\_a}^{\frac {2 b}{a}} a^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {2 b}{a}} b^{4} \operatorname {c0} +4 b \operatorname {c3} \,a^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 \operatorname {c3} \,a^{2} b^{2} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 a \operatorname {c3} \,b^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}-4 c_{1} b^{5}+16 a \operatorname {c2} \,b^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+11 \operatorname {c4} \,a^{2} b^{2} \textit {\_a}^{\frac {4 a +2 b}{a}}+12 a \operatorname {c4} \,b^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}+6 b \operatorname {c2} \,a^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+19 \operatorname {c2} \,a^{2} b^{2} \textit {\_a}^{\frac {2 a +2 b}{a}}+3 b \operatorname {c4} \,a^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}+35 \textit {\_a}^{\frac {2 b}{a}} a^{2} b^{2} \operatorname {c0} +20 \textit {\_a}^{\frac {2 b}{a}} a \,b^{3} \operatorname {c0} -25 c_{1} a^{3} b^{2}-6 c_{1} a^{4} b -20 c_{1} a \,b^{4}+4 \operatorname {c2} \,b^{4} \textit {\_a}^{\frac {2 a +2 b}{a}}+4 \operatorname {c4} \,b^{4} \textit {\_a}^{\frac {4 a +2 b}{a}}\right )}}d \textit {\_a} -x -c_{2} = 0 \int _{}^{y \left (x \right )}-\frac {\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 b \,a^{3}+35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right )}{\sqrt {-\textit {\_a}^{\frac {2 b}{a}} b \left (6 a^{4}+25 b \,a^{3}+35 b^{2} a^{2}+20 a \,b^{3}+4 b^{4}\right ) \left (25 \textit {\_a}^{\frac {2 b}{a}} a^{3} b \operatorname {c0} -35 c_{1} a^{2} b^{3}+4 \operatorname {c3} \,b^{4} \textit {\_a}^{\frac {3 a +2 b}{a}}+12 b \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{3}+26 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,a^{2} b^{2}+18 a \,\textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{3}+4 \textit {\_a}^{\frac {a +2 b}{a}} \operatorname {c1} \,b^{4}+6 \textit {\_a}^{\frac {2 b}{a}} a^{4} \operatorname {c0} +4 \textit {\_a}^{\frac {2 b}{a}} b^{4} \operatorname {c0} +4 b \operatorname {c3} \,a^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 \operatorname {c3} \,a^{2} b^{2} \textit {\_a}^{\frac {3 a +2 b}{a}}+14 a \operatorname {c3} \,b^{3} \textit {\_a}^{\frac {3 a +2 b}{a}}-4 c_{1} b^{5}+16 a \operatorname {c2} \,b^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+11 \operatorname {c4} \,a^{2} b^{2} \textit {\_a}^{\frac {4 a +2 b}{a}}+12 a \operatorname {c4} \,b^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}+6 b \operatorname {c2} \,a^{3} \textit {\_a}^{\frac {2 a +2 b}{a}}+19 \operatorname {c2} \,a^{2} b^{2} \textit {\_a}^{\frac {2 a +2 b}{a}}+3 b \operatorname {c4} \,a^{3} \textit {\_a}^{\frac {4 a +2 b}{a}}+35 \textit {\_a}^{\frac {2 b}{a}} a^{2} b^{2} \operatorname {c0} +20 \textit {\_a}^{\frac {2 b}{a}} a \,b^{3} \operatorname {c0} -25 c_{1} a^{3} b^{2}-6 c_{1} a^{4} b -20 c_{1} a \,b^{4}+4 \operatorname {c2} \,b^{4} \textit {\_a}^{\frac {2 a +2 b}{a}}+4 \operatorname {c4} \,b^{4} \textit {\_a}^{\frac {4 a +2 b}{a}}\right )}}d \textit {\_a} -x -c_{2} = 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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