60.7.156 problem 1747 (book 6.156)

Internal problem ID [11745]
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 : Tuesday, January 28, 2025 at 06:11:17 PM
CAS classification : [NONE]

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

✓ Solution by Maple

Time used: 0.240 (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 &= \operatorname {RootOf}\left (-2 b \left (\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} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )\right ) \left (a \,x^{2}+b x +c \right )^{{3}/{2}} \\ y &= \operatorname {RootOf}\left (2 b \left (\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} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}-2 b \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )\right ) \left (a \,x^{2}+b x +c \right )^{{3}/{2}} \\ \end{align*}

✓ Solution by Mathematica

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

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

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