60.7.137 problem 1728 (book 6.137)
Internal
problem
ID
[11726]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1728
(book
6.137)
Date
solved
:
Monday, January 27, 2025 at 11:31:53 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} 2 y^{\prime \prime } y+{y^{\prime }}^{2}+1&=0 \end{align*}
✓ Solution by Maple
Time used: 0.085 (sec). Leaf size: 327
dsolve(2*diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2+1=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (-\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} -2 c_{2} -2 x \right )\right ) c_{1} +2 x +2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} -2 c_{2} -2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\
y &= \frac {\left (-\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} +2 c_{2} +2 x \right )\right ) c_{1} -2 x -2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} +2 c_{2} +2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\
y &= \frac {\left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} -2 c_{2} -2 x \right )\right ) c_{1} -2 x -2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (-\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} -2 c_{2} -2 x \right ) \left (\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} -2 c_{2} -2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\
y &= \frac {\left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} +2 c_{2} +2 x \right )\right ) c_{1} +2 x +2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\left (\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} +2 c_{2} +2 x \right ) \left (-\cos \left (\textit {\_Z} \right ) c_{1} +\textit {\_Z} c_{1} +2 c_{2} +2 x \right )\right )\right )}{2}+\frac {c_{1}}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.360 (sec). Leaf size: 516
DSolve[1 + D[y[x],x]^2 + 2*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-c_1}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-c_1}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {e^{3 (-c_1)} \sqrt {1-\text {$\#$1} e^{-2 (-c_1)}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-(-c_1)}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 (-c_1)}\right )}{\sqrt {-\text {$\#$1}+e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {e^{3 (-c_1)} \sqrt {1-\text {$\#$1} e^{-2 (-c_1)}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-(-c_1)}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 (-c_1)}\right )}{\sqrt {-\text {$\#$1}+e^{2 (-c_1)}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-c_1}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {e^{3 c_1} \sqrt {1-\text {$\#$1} e^{-2 c_1}} \arcsin \left (\sqrt {\text {$\#$1}} e^{-c_1}\right )+\sqrt {\text {$\#$1}} \left (\text {$\#$1}-e^{2 c_1}\right )}{\sqrt {-\text {$\#$1}+e^{2 c_1}}}\&\right ][x+c_2] \\
\end{align*}