Internal problem ID [10114]
Internal file name [OUTPUT/9061_Monday_June_06_2022_06_19_13_AM_92348365/index.tex
]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1793 (book 6.202).
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.
\[ \boxed {a y \left (-1+y\right ) y^{\prime \prime }+\left (b y+c \right ) {y^{\prime }}^{2}+h \left (y\right )=0} \]
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)+(_b(_a)^2*_a*b+_b(_a)^2*c+h(_a))/(a*_a*(_a-1)) = 0, _b(_a)` *** Sublev 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: 166
dsolve(a*y(x)*(-1+y(x))*diff(diff(y(x),x),x)+(b*y(x)+c)*diff(y(x),x)^2+h(y(x))=0,y(x), singsol=all)
\begin{align*} a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_b}^{-\frac {c}{a}} \left (\textit {\_b} -1\right )^{\frac {c +b}{a}}}{\sqrt {-2 a \left (-\frac {c_{1} a}{2}+\int \textit {\_b}^{\frac {-a -2 c}{a}} h \left (\textit {\_b} \right ) \left (\textit {\_b} -1\right )^{\frac {2 c +2 b -a}{a}}d \textit {\_b} \right )}}d \textit {\_b} \right )-x -c_{2} &= 0 \\ -a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_b}^{-\frac {c}{a}} \left (\textit {\_b} -1\right )^{\frac {c +b}{a}}}{\sqrt {-2 a \left (-\frac {c_{1} a}{2}+\int \textit {\_b}^{\frac {-a -2 c}{a}} h \left (\textit {\_b} \right ) \left (\textit {\_b} -1\right )^{\frac {2 c +2 b -a}{a}}d \textit {\_b} \right )}}d \textit {\_b} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.75 (sec). Leaf size: 698
DSolve[h[y[x]] + (c + b*y[x])*y'[x]^2 + a*(-1 + y[x])*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {(1-K[2])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[2]^{-\frac {c}{a}}}{\sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {c \log (K[1])}{a}-\frac {(b+c) \log (1-K[1])}{a}\right )\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(1-K[3])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[3]^{-\frac {c}{a}}}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {c \log (K[1])}{a}-\frac {(b+c) \log (1-K[1])}{a}\right )\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {(1-K[2])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[2]^{-\frac {c}{a}}}{\sqrt {2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {c \log (K[1])}{a}-\frac {(b+c) \log (1-K[1])}{a}\right )\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]-c_1}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {(1-K[2])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[2]^{-\frac {c}{a}}}{\sqrt {c_1+2 \int _1^{K[2]}-\frac {\exp \left (-2 \left (\frac {c \log (K[1])}{a}-\frac {(b+c) \log (1-K[1])}{a}\right )\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(1-K[3])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[3]^{-\frac {c}{a}}}{\sqrt {2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {c \log (K[1])}{a}-\frac {(b+c) \log (1-K[1])}{a}\right )\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]-c_1}}dK[3]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(1-K[3])^{\frac {1}{2} \left (\frac {2 b}{a}+\frac {2 c}{a}\right )} K[3]^{-\frac {c}{a}}}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \left (\frac {c \log (K[1])}{a}-\frac {(b+c) \log (1-K[1])}{a}\right )\right ) h(K[1])}{a (K[1]-1) K[1]}dK[1]}}dK[3]\&\right ][x+c_2] \\ \end{align*}