Internal problem ID [10073]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1751 (book 6.160).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {4 y^{\prime \prime } y-3 {y^{\prime }}^{2}+a y^{3}+b y^{2}+y c=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 90
dsolve(4*diff(diff(y(x),x),x)*y(x)-3*diff(y(x),x)^2+a*y(x)^3+y(x)^2*b+c*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ -\sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (-a \,\textit {\_a}^{2}-3 \textit {\_a} b +3 c_{1} \sqrt {\textit {\_a}}+3 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} \left (-a \,\textit {\_a}^{2}-3 \textit {\_a} b +3 c_{1} \sqrt {\textit {\_a}}+3 c \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.052 (sec). Leaf size: 323
DSolve[c*y[x] + b*y[x]^2 + a*y[x]^3 - 3*y'[x]^2 + 4*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2+c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2+c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2-c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{\sqrt {-\frac {1}{3} a K[1]^3-b K[1]^2+c_1 K[1]^{3/2}+c K[1]}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2-c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {-\frac {1}{3} a K[2]^3-b K[2]^2+c_1 K[2]^{3/2}+c K[2]}}dK[2]\&\right ][x+c_2] \\ \end{align*}