58.1.19 problem 19
Internal
problem
ID
[9090]
Book
:
Second
order
enumerated
odes
Section
:
section
1
Problem
number
:
19
Date
solved
:
Monday, January 27, 2025 at 05:37:15 PM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} {y^{\prime \prime }}^{2}+y^{\prime }&=x \end{align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 122
dsolve(diff(y(x),x$2)^2+diff(y(x),x)=x,y(x), singsol=all)
\begin{align*}
y &= \int \left (-{\mathrm e}^{2 \operatorname {RootOf}\left (\textit {\_Z} -x -2 \,{\mathrm e}^{\textit {\_Z}}+2+c_{1} -\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}+2 \,{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z} -x -2 \,{\mathrm e}^{\textit {\_Z}}+2+c_{1} -\ln \left ({\mathrm e}^{\textit {\_Z}} \left ({\mathrm e}^{\textit {\_Z}}-2\right )^{2}\right )\right )}+x \right )d x -x +c_{2} \\
y &= \frac {2 \operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )^{3}}{3}+3 \operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )^{2}+4 \operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-\frac {x}{2}-1}\right )+\frac {x^{2}}{2}-x +c_{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 17.332 (sec). Leaf size: 172
DSolve[(D[y[x],{x,2}])^2+D[y[x],x]==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2}{3} W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^3+3 W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^2+4 W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right )+\frac {x^2}{2}-x+c_2 \\
y(x)\to \frac {2}{3} W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right ){}^3+3 W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right ){}^2+4 W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right )+\frac {x^2}{2}-x+c_2 \\
y(x)\to \frac {x^2}{2}-x+c_2 \\
\end{align*}