Internal problem ID [10112]
Internal file name [OUTPUT/9059_Monday_June_06_2022_06_19_01_AM_76780109/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1791 (book 6.200).
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, _missing_x]]
Unable to solve or complete the solution.
\[ \boxed {3 y \left (1-y\right ) y^{\prime \prime }-2 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right )=0} \]
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 `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)+(1/3)*(-4*_a*_b(_a)^2+2*_b(_a)^2+h(_a))/(_a*(_a-1)) = 0, _b(_a)` *** S Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 120
dsolve(3*y(x)*(1-y(x))*diff(diff(y(x),x),x)-2*(1-2*y(x))*diff(y(x),x)^2-h(y(x))=0,y(x), singsol=all)
\begin{align*} -\sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} \textit {\_b} \left (\textit {\_b} -1\right ) \left (2 \left (\int \frac {h \left (\textit {\_b} \right )}{\textit {\_b}^{2} \left (\textit {\_b} -1\right )^{2} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}}}d \textit {\_b} \right )-3 c_{1} \right )}}d \textit {\_b} \right )-x -c_{2} &= 0 \\ \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}} \textit {\_b} \left (\textit {\_b} -1\right ) \left (2 \left (\int \frac {h \left (\textit {\_b} \right )}{\textit {\_b}^{2} \left (\textit {\_b} -1\right )^{2} \left (\textit {\_b} \left (\textit {\_b} -1\right )\right )^{\frac {1}{3}}}d \textit {\_b} \right )-3 c_{1} \right )}}d \textit {\_b} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.69 (sec). Leaf size: 560
DSolve[-h[y[x]] - 2*(1 - 2*y[x])*y'[x]^2 + 3*(1 - y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(1-K[2])^{2/3} K[2]^{2/3} \sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(1-K[3])^{2/3} K[3]^{2/3} \sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(1-K[2])^{2/3} K[2]^{2/3} \sqrt {2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(1-K[2])^{2/3} K[2]^{2/3} \sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(1-K[3])^{2/3} K[3]^{2/3} \sqrt {2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(1-K[3])^{2/3} K[3]^{2/3} \sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {2}{3} \log (1-K[1])+\frac {2}{3} \log (K[1])\right )\right ) h(K[1])}{3 (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ \end{align*}