60.7.132 problem 1747 (book 6.156)

Internal problem ID [11682]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1747 (book 6.156)
Date solved : Sunday, March 30, 2025 at 08:41:56 PM
CAS classification : [NONE]

\begin{align*} 3 y^{\prime \prime } y-2 {y^{\prime }}^{2}-a \,x^{2}-b x -c&=0 \end{align*}

✓ Maple. Time used: 0.143 (sec). Leaf size: 205

ode:=3*diff(diff(y(x),x),x)*y(x)-2*diff(y(x),x)^2-a*x^2-b*x-c = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (-2 b \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 \textit {\_f}^{{4}/{3}} c_1 \,b^{2}-36 \textit {\_f}^{2} a c +9 \textit {\_f}^{2} b^{2}-2}}d \textit {\_f} \sqrt {4 a c -b^{2}}-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+c_2 \sqrt {4 a c -b^{2}}\right ) \left (a \,x^{2}+b x +c \right )^{{3}/{2}} \\ y &= \operatorname {RootOf}\left (2 b \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 \textit {\_f}^{{4}/{3}} c_1 \,b^{2}-36 \textit {\_f}^{2} a c +9 \textit {\_f}^{2} b^{2}-2}}d \textit {\_f} \sqrt {4 a c -b^{2}}-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )+c_2 \sqrt {4 a c -b^{2}}\right ) \left (a \,x^{2}+b x +c \right )^{{3}/{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.097 (sec). Leaf size: 118

ode=-c - b*x - a*x^2 - 2*D[y[x],x]^2 + 3*y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int \frac {y(x)^{2/3}}{\left (a x^2+b x+c\right ) \sqrt {-\frac {2 \left (a x^2+b x+c\right )^3}{y(x)^2}+\frac {c_1 \left (a x^2+b x+c\right )}{y(x)^{2/3}}+9 \left (b^2-4 a c\right )}}d\frac {a x^2+b x+c}{y(x)^{2/3}}=-\int \frac {1}{3 \left (a x^2+b x+c\right )}dx+c_2,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-a*x**2 - b*x - c + 3*y(x)*Derivative(y(x), (x, 2)) - 2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(2)*sqrt(-a*x**2 - b*x - c + 3*y(x)*Derivative(y(x), (x, 2)))/2 + Derivative(y(x), x) cannot be solved by the factorable group method