60.7.30 problem 1620 (6.30)
Internal
problem
ID
[11619]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1620
(6.30)
Date
solved
:
Monday, January 27, 2025 at 11:27:16 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{\prime \prime }+y^{\prime } y-y^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 332
dsolve(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3=0,y(x), singsol=all)
\begin{align*}
2 \left (\int _{}^{y}\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}}{\textit {\_a}^{4}-\textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
-4 \left (\int _{}^{y}\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}}{-i \sqrt {3}\, \textit {\_a}^{4}+i \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}} \sqrt {3}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
4 \left (\int _{}^{y}-\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}}{i \sqrt {3}\, \textit {\_a}^{4}-i \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}} \sqrt {3}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{1}/{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{{2}/{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 52.756 (sec). Leaf size: 1534
DSolve[-y[x]^3 + y[x]*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 {$\#$1}}\frac {2}{\frac {e^{6 c_1} K[1]^4}{\sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}-K[1]^2+e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1} K[2]^4}{4 \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}-\frac {K[2]^2}{2}-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 c_1} K[3]^4}{4 \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}-\frac {K[3]^2}{2}-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2}{\frac {e^{6 (-c_1)} K[1]^4}{\sqrt [3]{e^{18 (-c_1)} K[1]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[1]^6}}}-K[1]^2+e^{-6 (-c_1)} \sqrt [3]{e^{18 (-c_1)} K[1]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[1]^6}}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2}{\frac {e^{6 c_1} K[1]^4}{\sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}-K[1]^2+e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[1]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[1]^6}}}dK[1]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 (-c_1)} K[2]^4}{4 \sqrt [3]{e^{18 (-c_1)} K[2]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[2]^6}}}-\frac {K[2]^2}{2}-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 (-c_1)} \sqrt [3]{e^{18 (-c_1)} K[2]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[2]^6}}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1} K[2]^4}{4 \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}-\frac {K[2]^2}{2}-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[2]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[2]^6}}}dK[2]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 (-c_1)} K[3]^4}{4 \sqrt [3]{e^{18 (-c_1)} K[3]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[3]^6}}}-\frac {K[3]^2}{2}-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 (-c_1)} \sqrt [3]{e^{18 (-c_1)} K[3]^6-2 e^{12 (-c_1)}+2 \sqrt {e^{24 (-c_1)}-e^{30 (-c_1)} K[3]^6}}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 c_1} K[3]^4}{4 \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}-\frac {K[3]^2}{2}-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[3]^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{30 c_1} K[3]^6}}}dK[3]\&\right ][x+c_2] \\
\end{align*}