Internal problem ID [10113]
Internal file name [OUTPUT/9060_Monday_June_06_2022_06_19_07_AM_35403722/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1792 (book 6.201).
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.
\[ \boxed {\left (1-y\right ) y^{\prime \prime }-3 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right )=0} \]
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)+(-6*_a*_b(_a)^2+3*_b(_a)^2+h(_a))/(_a-1) = 0, _b(_a)` *** Sublevel 2 * 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: 0.015 (sec). Leaf size: 91
dsolve((1-y(x))*diff(diff(y(x),x),x)-3*(1-2*y(x))*diff(y(x),x)^2-h(y(x))=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {{\mathrm e}^{-6 \textit {\_b}}}{\sqrt {-2 \left (\int \frac {{\mathrm e}^{-12 \textit {\_b}} h \left (\textit {\_b} \right )}{\left (\textit {\_b} -1\right )^{7}}d \textit {\_b} \right )+c_{1}}\, \left (\textit {\_b} -1\right )^{3}}d \textit {\_b} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {{\mathrm e}^{-6 \textit {\_b}}}{\sqrt {-2 \left (\int \frac {{\mathrm e}^{-12 \textit {\_b}} h \left (\textit {\_b} \right )}{\left (\textit {\_b} -1\right )^{7}}d \textit {\_b} \right )+c_{1}}\, \left (\textit {\_b} -1\right )^{3}}d \textit {\_b} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.515 (sec). Leaf size: 506
DSolve[-h[y[x]] - 3*(1 - 2*y[x])*y'[x]^2 + (1 - y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{\frac {1}{2} (12-12 K[2])}}{(K[2]-1)^3 \sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp (-2 (6 (K[1]-1)+3 \log (K[1]-1))) h(K[1])}{K[1]-1}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{\frac {1}{2} (12-12 K[3])}}{(K[3]-1)^3 \sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp (-2 (6 (K[1]-1)+3 \log (K[1]-1))) h(K[1])}{K[1]-1}dK[1]}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{\frac {1}{2} (12-12 K[2])}}{(K[2]-1)^3 \sqrt {2 \int _1^{K[2]}-\frac {\exp (-2 (6 (K[1]-1)+3 \log (K[1]-1))) h(K[1])}{K[1]-1}dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{\frac {1}{2} (12-12 K[2])}}{(K[2]-1)^3 \sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp (-2 (6 (K[1]-1)+3 \log (K[1]-1))) h(K[1])}{K[1]-1}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{\frac {1}{2} (12-12 K[3])}}{(K[3]-1)^3 \sqrt {2 \int _1^{K[3]}-\frac {\exp (-2 (6 (K[1]-1)+3 \log (K[1]-1))) h(K[1])}{K[1]-1}dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{\frac {1}{2} (12-12 K[3])}}{(K[3]-1)^3 \sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp (-2 (6 (K[1]-1)+3 \log (K[1]-1))) h(K[1])}{K[1]-1}dK[1]}}dK[3]\&\right ][x+c_2] \\ \end{align*}