60.7.154 problem 1745 (book 6.154)
Internal
problem
ID
[11743]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1745
(book
6.154)
Date
solved
:
Monday, January 27, 2025 at 11:32:19 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} 2 y^{\prime \prime } y-{y^{\prime }}^{2} \left ({y^{\prime }}^{2}+1\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.102 (sec). Leaf size: 331
dsolve(2*diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2*(diff(y(x),x)^2+1)=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
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: 0.307 (sec). Leaf size: 51
DSolve[-(D[y[x],x]^2*(1 + D[y[x],x]^2)) + 2*y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (K[1]^2+1\right )}dK[1]\&\right ]\left [c_1+\frac {1}{2} \log (K[2])\right ]}dK[2]=x+c_2,y(x)\right ]
\]