Internal problem ID [9330]
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]]
\[ \boxed {a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0}=0} \]
✓ 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: 4.411 (sec). Leaf size: 716
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]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {4 b^5+20 a b^4+35 a^2 b^3+25 a^3 b^2+6 a^4 b}}{\sqrt {4 b^5 c_1 K[1]^{-\frac {2 b}{a}}+20 a b^4 c_1 K[1]^{-\frac {2 b}{a}}+35 a^2 b^3 c_1 K[1]^{-\frac {2 b}{a}}+25 a^3 b^2 c_1 K[1]^{-\frac {2 b}{a}}+6 a^4 b c_1 K[1]^{-\frac {2 b}{a}}-4 b^4 \text {c4} K[1]^4-12 a b^3 \text {c4} K[1]^4-11 a^2 b^2 \text {c4} K[1]^4-3 a^3 b \text {c4} K[1]^4-4 b^4 \text {c3} K[1]^3-14 a b^3 \text {c3} K[1]^3-14 a^2 b^2 \text {c3} K[1]^3-4 a^3 b \text {c3} K[1]^3-4 b^4 \text {c2} K[1]^2-16 a b^3 \text {c2} K[1]^2-19 a^2 b^2 \text {c2} K[1]^2-6 a^3 b \text {c2} K[1]^2-4 b^4 \text {c1} K[1]-18 a b^3 \text {c1} K[1]-26 a^2 b^2 \text {c1} K[1]-12 a^3 b \text {c1} K[1]-6 a^4 \text {c0}-4 b^4 \text {c0}-20 a b^3 \text {c0}-35 a^2 b^2 \text {c0}-25 a^3 b \text {c0}}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {4 b^5+20 a b^4+35 a^2 b^3+25 a^3 b^2+6 a^4 b}}{\sqrt {4 b^5 c_1 K[2]^{-\frac {2 b}{a}}+20 a b^4 c_1 K[2]^{-\frac {2 b}{a}}+35 a^2 b^3 c_1 K[2]^{-\frac {2 b}{a}}+25 a^3 b^2 c_1 K[2]^{-\frac {2 b}{a}}+6 a^4 b c_1 K[2]^{-\frac {2 b}{a}}-4 b^4 \text {c4} K[2]^4-12 a b^3 \text {c4} K[2]^4-11 a^2 b^2 \text {c4} K[2]^4-3 a^3 b \text {c4} K[2]^4-4 b^4 \text {c3} K[2]^3-14 a b^3 \text {c3} K[2]^3-14 a^2 b^2 \text {c3} K[2]^3-4 a^3 b \text {c3} K[2]^3-4 b^4 \text {c2} K[2]^2-16 a b^3 \text {c2} K[2]^2-19 a^2 b^2 \text {c2} K[2]^2-6 a^3 b \text {c2} K[2]^2-4 b^4 \text {c1} K[2]-18 a b^3 \text {c1} K[2]-26 a^2 b^2 \text {c1} K[2]-12 a^3 b \text {c1} K[2]-6 a^4 \text {c0}-4 b^4 \text {c0}-20 a b^3 \text {c0}-35 a^2 b^2 \text {c0}-25 a^3 b \text {c0}}}dK[2]\&\right ][x+c_2] \\ \end{align*}