Internal problem ID [10068]
Internal file name [OUTPUT/9015_Monday_June_06_2022_06_13_15_AM_6778138/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {3 y^{\prime \prime } y-2 {y^{\prime }}^{2}=a \,x^{2}+b x +c} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying symmetries linear in x and y(x) trying differential order: 2; exact nonlinear trying 2nd order, integrating factor of the form mu(y) trying 2nd order, integrating factor of the form mu(x,y) trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case trying 2nd order, integrating factor of the form mu(y,y) trying differential order: 2; mu polynomial in y trying 2nd order, integrating factor of the form mu(x,y) differential order: 2; looking for linear symmetries -> trying 2nd order, the S-function method -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for the S-function -> trying 2nd order, the S-function method -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^ --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)* --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[(a*x^2+b*x+c)/b, 3/2*(2*a*x+b)*y/b]
✓ 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]
\[ \text {Solve}\left [\int \frac {y(x)^{2/3}}{\left (a x^2+b x+c\right ) \sqrt {-\frac {2 \left (a x^2+b x+c\right )^3}{y(x)^2}+\frac {c_1 \left (a x^2+b x+c\right )}{y(x)^{2/3}}+9 \left (b^2-4 a c\right )}}d\frac {a x^2+b x+c}{y(x)^{2/3}}=-\int \frac {1}{3 \left (a x^2+b x+c\right )}dx+c_2,y(x)\right ] \]