60.7.202 problem 1793 (book 6.202)
Internal
problem
ID
[11791]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1793
(book
6.202)
Date
solved
:
Monday, January 27, 2025 at 11:35:52 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} a y \left (y-1\right ) y^{\prime \prime }+\left (b y+c \right ) {y^{\prime }}^{2}+h \left (y\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.026 (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}\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}\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: 1.396 (sec). Leaf size: 752
DSolve[h[y[x]] + (c + b*y[x])*D[y[x],x]^2 + a*(-1 + y[x])*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{a (K[2]-1) K[2]}dK[2]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[4]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{a (K[2]-1) K[2]}dK[2]}}dK[4]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right )}{\sqrt {2 \int _1^{K[3]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{a (K[2]-1) K[2]}dK[2]-c_1}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{a (K[2]-1) K[2]}dK[2]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right )}{\sqrt {2 \int _1^{K[4]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{a (K[2]-1) K[2]}dK[2]-c_1}}dK[4]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[4]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {-c-b K[1]}{a (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{a (K[2]-1) K[2]}dK[2]}}dK[4]\&\right ][x+c_2] \\
\end{align*}