Internal problem ID [10069]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1747 (book 6.156).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [NONE]
\[ \boxed {3 y^{\prime \prime } y-2 {y^{\prime }}^{2}=a \,x^{2}+b x +c} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 205
dsolve(3*diff(diff(y(x),x),x)*y(x)-2*diff(y(x),x)^2-a*x^2-b*x-c=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )-2 b \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 \textit {\_f}^{\frac {4}{3}} c_{1} b^{2}-36 c \,\textit {\_f}^{2} a +9 b^{2} \textit {\_f}^{2}-2}}d \textit {\_f} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}\right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+2 b \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 \textit {\_f}^{\frac {4}{3}} c_{1} b^{2}-36 c \,\textit {\_f}^{2} a +9 b^{2} \textit {\_f}^{2}-2}}d \textit {\_f} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}\right ) \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.129 (sec). Leaf size: 118
DSolve[-c - b*x - a*x^2 - 2*y'[x]^2 + 3*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int \frac {y(x)^{2/3}}{\left (a x^2+b x+c\right ) \sqrt {-\frac {2 \left (a x^2+b x+c\right )^3}{y(x)^2}+\frac {c_1 \left (a x^2+b x+c\right )}{y(x)^{2/3}}+9 \left (b^2-4 a c\right )}}d\frac {a x^2+b x+c}{y(x)^{2/3}}=-\int \frac {1}{3 \left (a x^2+b x+c\right )}dx+c_2,y(x)\right ] \]