58.1.46 problem 46
Internal
problem
ID
[9117]
Book
:
Second
order
enumerated
odes
Section
:
section
1
Problem
number
:
46
Date
solved
:
Monday, January 27, 2025 at 05:42:41 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y {y^{\prime \prime }}^{2}+{y^{\prime }}^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.161 (sec). Leaf size: 166
dsolve(y(x)*diff(y(x),x$2)^2+diff(y(x),x)^3=0,y(x), singsol=all)
\begin{align*}
y &= c_{1} \\
y &= 0 \\
y &= \frac {c_{2} {\left (\operatorname {LambertW}\left (c_{1} {\mathrm e}^{\frac {x}{2}-1}\right )+1\right )}^{2}}{\operatorname {LambertW}\left (c_{1} {\mathrm e}^{\frac {x}{2}-1}\right )^{2}} \\
y &= \frac {c_{2} {\left (\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{\frac {x}{2}-1}\right )+1\right )}^{2}}{\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{\frac {x}{2}-1}\right )^{2}} \\
y &= {\mathrm e}^{-\int {\mathrm e}^{2 \operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} {\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{\textit {\_Z}}+\ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} -2 \textit {\_Z} +x -2\right )}d x -2 \left (\int {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} {\mathrm e}^{\textit {\_Z}}-2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+x \,{\mathrm e}^{\textit {\_Z}}+\ln \left (\left ({\mathrm e}^{\textit {\_Z}}+1\right )^{2}\right )+c_{1} -2 \textit {\_Z} +x -2\right )}d x \right )-x +c_{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.974 (sec). Leaf size: 361
DSolve[y[x]*D[y[x],{x,2}]^2+D[y[x],x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i c_1\right )-\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}-i c_1\right )}\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}+i c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i c_1\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i (-c_1)\right )-\frac {i (-c_1)}{2 \left (2 \sqrt {\text {$\#$1}}-i (-c_1)\right )}\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i (-c_1)}{2 \left (2 \sqrt {\text {$\#$1}}+i (-1) c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i (-1) c_1\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-4 \left (\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}-i c_1\right )-\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}-i c_1\right )}\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-4 \left (\frac {i c_1}{2 \left (2 \sqrt {\text {$\#$1}}+i c_1\right )}+\frac {1}{2} \log \left (2 \sqrt {\text {$\#$1}}+i c_1\right )\right )\&\right ][x+c_2] \\
\end{align*}