Internal problem ID [10013]
Internal file name [OUTPUT/8960_Monday_June_06_2022_06_03_34_AM_19665931/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1692 (book 6.101).
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.
\[ \boxed {\left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} y^{\prime \prime }-F \left (\frac {y}{\sqrt {a \,x^{2}+b x +c}}\right )=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- 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 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+b*x+c)/a, 1/2*y*(2*a*x+b)/a]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 252
dsolve((a*x^2+b*x+c)^(3/2)*diff(diff(y(x),x),x)-F(y(x)/(a*x^2+b*x+c)^(1/2))=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (4 \textit {\_Z} a c -\textit {\_Z} \,b^{2}-4 F \left (\frac {\textit {\_Z}}{\sqrt {a \,x^{2}+b x +c}}\right ) \sqrt {a \,x^{2}+b x +c}\right ) \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right ) a -2 a \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 c_{1} a^{2}-4 c \,\textit {\_g}^{2} a +b^{2} \textit {\_g}^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}\right ) \sqrt {a \,x^{2}+b x +c} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right ) a +2 a \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {4 c_{1} a^{2}-4 c \,\textit {\_g}^{2} a +b^{2} \textit {\_g}^{2}+8 \left (\int F \left (\textit {\_g} \right )d \textit {\_g} \right )}}d \textit {\_g} \right ) \sqrt {4 a c -b^{2}}+c_{2} \sqrt {4 a c -b^{2}}\right ) \sqrt {a \,x^{2}+b x +c} \\ \end{align*}
✓ Solution by Mathematica
Time used: 55.307 (sec). Leaf size: 251
DSolve[-f[y[x]/Sqrt[c + b*x + a*x^2]] + (c + b*x + a*x^2)^(3/2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [2 a \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )+2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[3]^2+8 \int _1^{K[3]}f(K[2])dK[2]}}dK[3]&=c_2 \sqrt {4 a c-b^2},y(x)\right ] \\ \text {Solve}\left [2 a \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )-2 \sqrt {4 a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (b+a x)}}}\frac {a}{\sqrt {4 c_1 a^2+\left (b^2-4 a c\right ) K[5]^2+8 \int _1^{K[5]}f(K[4])dK[4]}}dK[5]&=c_2 \sqrt {4 a c-b^2},y(x)\right ] \\ \end{align*}