Internal problem ID [10131]
Internal file name [OUTPUT/9078_Monday_June_06_2022_06_24_53_AM_81305351/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1810 (book 6.219).
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.
\[ \boxed {\left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+y d=0} \]
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 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 2nd order, No Point Symmetries Class V -> 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 Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3`[(a*x^2+2*b*x+c)/a, y*(a*x+b)/a]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 336
dsolve((c+2*b*x+a*x^2+y(x)^2)^2*diff(diff(y(x),x),x)+d*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (a \left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_f}^{2} a^{2} c_{1} -c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+a^{2} c_{1} +d \right )}}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_f}^{2} a^{2} c_{1} -c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+a^{2} c_{1} +d}d \textit {\_f} \right ) \sqrt {a c -b^{2}}-a \arctan \left (\frac {a x +b}{\sqrt {a c -b^{2}}}\right )+c_{2} \sqrt {a c -b^{2}}\right ) \sqrt {a \,x^{2}+2 b x +c} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-a \left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_f}^{2} a^{2} c_{1} -c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+a^{2} c_{1} +d \right )}}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+\textit {\_f}^{2} a^{2} c_{1} -c \,\textit {\_f}^{2} a +b^{2} \textit {\_f}^{2}+a^{2} c_{1} +d}d \textit {\_f} \right ) \sqrt {a c -b^{2}}-a \arctan \left (\frac {a x +b}{\sqrt {a c -b^{2}}}\right )+c_{2} \sqrt {a c -b^{2}}\right ) \sqrt {a \,x^{2}+2 b x +c} \\ \end{align*}
✓ Solution by Mathematica
Time used: 65.538 (sec). Leaf size: 260
DSolve[d*y[x] + (c + 2*b*x + a*x^2 + y[x]^2)^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [a \arctan \left (\frac {a x+b}{\sqrt {a c-b^2}}\right )+\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[2]^2+1\right )}{\sqrt {\left (K[2]^2+1\right ) \left (d+\left (K[2]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[2]^2\right )\right )}}dK[2]&=c_2 \sqrt {a c-b^2},y(x)\right ] \\ \text {Solve}\left [a \arctan \left (\frac {a x+b}{\sqrt {a c-b^2}}\right )-\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[3]^2+1\right )}{\sqrt {\left (K[3]^2+1\right ) \left (d+\left (K[3]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[3]^2\right )\right )}}dK[3]&=c_2 \sqrt {a c-b^2},y(x)\right ] \\ \end{align*}