7.197 problem 1788 (book 6.197)

Internal problem ID [10109]
Internal file name [OUTPUT/9056_Monday_June_06_2022_06_18_39_AM_72378699/index.tex]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1788 (book 6.197).
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.

\[ \boxed {2 y \left (1-y\right ) y^{\prime \prime }-\left (1-3 y\right ) {y^{\prime }}^{2}+h \left (y\right )=0} \]

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)-(1/2)*(3*_a*_b(_a)^2-_b(_a)^2+h(_a))/(_a*(_a-1)) = 0, _b(_a)`   *** Subl 
   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: 0.016 (sec). Leaf size: 81

dsolve(2*y(x)*(1-y(x))*diff(diff(y(x),x),x)-(1-3*y(x))*diff(y(x),x)^2+h(y(x))=0,y(x), singsol=all)
 

\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_b} \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b} -1\right )^{3} \textit {\_b}^{2}}d \textit {\_b} +c_{1} \right )}\, \left (\textit {\_b} -1\right )}d \textit {\_b} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_b} \left (\int \frac {h \left (\textit {\_b} \right )}{\left (\textit {\_b} -1\right )^{3} \textit {\_b}^{2}}d \textit {\_b} +c_{1} \right )}\, \left (\textit {\_b} -1\right )}d \textit {\_b} \right )-x -c_{2} &= 0 \\ \end{align*}

Solution by Mathematica

Time used: 1.941 (sec). Leaf size: 512

DSolve[h[y[x]] - (1 - 3*y[x])*y'[x]^2 + 2*(1 - y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(K[2]-1) \sqrt {K[2]} \sqrt {c_1+2 \int _1^{K[2]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[3]-1) \sqrt {K[3]} \sqrt {c_1+2 \int _1^{K[3]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(K[2]-1) \sqrt {K[2]} \sqrt {2 \int _1^{K[2]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {1}{(K[2]-1) \sqrt {K[2]} \sqrt {c_1+2 \int _1^{K[2]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[3]-1) \sqrt {K[3]} \sqrt {2 \int _1^{K[3]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[3]-1) \sqrt {K[3]} \sqrt {c_1+2 \int _1^{K[3]}\frac {e^{-2 \left (\log (1-K[1])+\frac {1}{2} \log (K[1])\right )} h(K[1])}{2 (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ \end{align*}