Internal problem ID [10145]
Internal file name [OUTPUT/9092_Monday_June_06_2022_06_30_35_AM_50293028/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1824 (book 6.233).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\left ({y^{\prime }}^{2}+a \left (x y^{\prime }-y\right )\right ) y^{\prime \prime }=b} \]
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 differential order: 2; found: 2 linear symmetries. Trying reduction of order `, `2nd order, trying reduction of order with given symmetries:`[1, -1/2*a*x], [0, -(_y1*a^2*x+_y1^2*a-a^2*y+2*b)/a]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 289
dsolve((diff(y(x),x)^2+a*(x*diff(y(x),x)-y(x)))*diff(diff(y(x),x),x)-b=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1} \right ) \left (a \textit {\_f} +\sqrt {4 \textit {\_f} b -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\ y \left (x \right ) &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1} \right ) \left (a \textit {\_f} -\sqrt {4 \textit {\_f} b -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1}}d \textit {\_f} +c_{2} \right ) \\ y \left (x \right ) &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x -\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1} \right ) \left (a \textit {\_f} +\sqrt {4 \textit {\_f} b -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1}}d \textit {\_f} \right )+c_{2} \right ) \\ y \left (x \right ) &= -\frac {a \,x^{2}}{4}+\operatorname {RootOf}\left (-x -\left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1} \right ) \left (a \textit {\_f} -\sqrt {4 \textit {\_f} b -2 c_{1}}\right )}}{\textit {\_f}^{2} a^{2}-4 \textit {\_f} b +2 c_{1}}d \textit {\_f} \right )+c_{2} \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.617 (sec). Leaf size: 281
DSolve[-b + (y'[x]^2 + a*(-y[x] + x*y'[x]))*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\int \frac {a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2-4 b \left (\frac {a x^2}{4}+y(x)\right )+2 c_1\right ) \left (a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )&=-x+c_2,y(x)\right ] \\ \text {Solve}\left [\int \frac {a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}}{\sqrt {\left (a^2 \left (\frac {a x^2}{4}+y(x)\right )^2-4 b \left (\frac {a x^2}{4}+y(x)\right )+2 c_1\right ) \left (a \left (\frac {a x^2}{4}+y(x)\right )+\sqrt {4 b \left (\frac {a x^2}{4}+y(x)\right )-2 c_1}\right )}}d\left (\frac {a x^2}{4}+y(x)\right )&=-x+c_2,y(x)\right ] \\ \end{align*}