60.7.67 problem 1658 (book 6.67)
Internal
problem
ID
[11656]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1658
(book
6.67)
Date
solved
:
Monday, January 27, 2025 at 11:29:33 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{\prime \prime }+y^{3} y^{\prime }-y y^{\prime } \sqrt {y^{4}+4 y^{\prime }}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.142 (sec). Leaf size: 131
dsolve(diff(diff(y(x),x),x)+y(x)^3*diff(y(x),x)-y(x)*diff(y(x),x)*(y(x)^4+4*diff(y(x),x))^(1/2)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {2^{{2}/{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{{1}/{3}}}{4 c_{1} +3 x} \\
y &= -\frac {2^{{2}/{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{8 c_{1} +6 x} \\
y &= \frac {2^{{2}/{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{8 c_{1} +6 x} \\
y &= \tan \left (\frac {x +c_{2}}{c_{1}^{3}}\right ) \sqrt {\frac {1}{c_{1}^{2}}} \\
y &= \tanh \left (\frac {x +c_{2}}{c_{1}^{3}}\right ) \sqrt {\frac {1}{c_{1}^{2}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.839 (sec). Leaf size: 317
DSolve[y[x]^3*D[y[x],x] - y[x]*D[y[x],x]*Sqrt[y[x]^4 + 4*D[y[x],x]] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(\cosh (2 c_1)+\sinh (2 c_1)) \left (-K[1]^2+\cosh (2 c_1)+\sinh (2 c_1)\right )}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(\cosh (2 c_1)+\sinh (2 c_1)) \left (K[2]^2+\cosh (2 c_1)+\sinh (2 c_1)\right )}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(\cosh (2 (-c_1))+\sinh (2 (-c_1))) \left (-K[1]^2+\cosh (2 (-c_1))+\sinh (2 (-c_1))\right )}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(\cosh (2 (-c_1))+\sinh (2 (-c_1))) \left (K[2]^2+\cosh (2 (-c_1))+\sinh (2 (-c_1))\right )}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(\cosh (2 c_1)+\sinh (2 c_1)) \left (-K[1]^2+\cosh (2 c_1)+\sinh (2 c_1)\right )}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(\cosh (2 c_1)+\sinh (2 c_1)) \left (K[2]^2+\cosh (2 c_1)+\sinh (2 c_1)\right )}dK[2]\&\right ][x+c_2] \\
\end{align*}