52.1.19 problem 19

Internal problem ID [10390]
Book : Second order enumerated odes
Section : section 1
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 07:22:58 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} {y^{\prime \prime }}^{2}+y^{\prime }&=x \end{align*}

✓ Maple. Time used: 0.334 (sec). Leaf size: 122

ode:=diff(diff(y(x),x),x)^2+diff(y(x),x) = x; 
dsolve(ode,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}^{-1-\frac {x}{2}}\right )^{3}}{3}+3 \operatorname {LambertW}\left (-c_1 \,{\mathrm e}^{-1-\frac {x}{2}}\right )^{2}+4 \operatorname {LambertW}\left (-c_1 \,{\mathrm e}^{-1-\frac {x}{2}}\right )+\frac {x^{2}}{2}-x +c_2 \\ \end{align*}

✓ Mathematica. Time used: 60.087 (sec). Leaf size: 117

ode=(D[y[x],{x,2}])^2+D[y[x],x]==x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\left (-W\left (e^{-\frac {c_1}{2}-\frac {K[1]}{2}-1}\right ){}^2-2 W\left (e^{-\frac {c_1}{2}-\frac {K[1]}{2}-1}\right )+K[1]-1\right )dK[1]+c_2\\ y(x)&\to \int _1^x\left (-W\left (-e^{\frac {1}{2} (c_1-K[2]-2)}\right ){}^2-2 W\left (-e^{\frac {1}{2} (c_1-K[2]-2)}\right )+K[2]-1\right )dK[2]+c_2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + Derivative(y(x), x) + Derivative(y(x), (x, 2))**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : multiple generators [log(sqrt(-_X0 + x) + 1), sqrt(-_X0 + x)] 
No algorithms are implemented to solve equation C1 + x - 2*sqrt(-_X0 + x) + 2*log(sqrt(-_X0 + x) + 1)