60.7.197 problem 1788 (book 6.197)
Internal
problem
ID
[11786]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1788
(book
6.197)
Date
solved
:
Monday, January 27, 2025 at 11:35:45 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} 2 y \left (1-y\right ) y^{\prime \prime }-\left (1-3 y\right ) {y^{\prime }}^{2}+h \left (y\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.025 (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}\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 \\
-\int _{}^{y}\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 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.856 (sec). Leaf size: 710
DSolve[h[y[x]] - (1 - 3*y[x])*D[y[x],x]^2 + 2*(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 {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}\frac {\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{2 (K[2]-1) K[2]}dK[2]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[4]}\frac {\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{2 (K[2]-1) K[2]}dK[2]}}dK[4]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right )}{\sqrt {2 \int _1^{K[3]}\frac {\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{2 (K[2]-1) K[2]}dK[2]-c_1}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}\frac {\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{2 (K[2]-1) K[2]}dK[2]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right )}{\sqrt {2 \int _1^{K[4]}\frac {\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{2 (K[2]-1) K[2]}dK[2]-c_1}}dK[4]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {3 K[1]-1}{2 (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 {3 K[1]-1}{2 (K[1]-1) K[1]}dK[1]\right ) h(K[2])}{2 (K[2]-1) K[2]}dK[2]}}dK[4]\&\right ][x+c_2] \\
\end{align*}