29.27.29 problem 795

Internal problem ID [5379]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 27
Problem number : 795
Date solved : Sunday, March 30, 2025 at 08:03:59 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} {y^{\prime }}^{2}+2 \left (1-x \right ) y^{\prime }-2 x +2 y&=0 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 32

ode:=diff(y(x),x)^2+2*(1-x)*diff(y(x),x)-2*x+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{2}}{2}-\frac {\operatorname {LambertW}\left (-c_1 \,{\mathrm e}^{-x}\right )^{2}}{2}-\operatorname {LambertW}\left (-c_1 \,{\mathrm e}^{-x}\right ) \]

✓ Mathematica. Time used: 1.515 (sec). Leaf size: 171

ode=(D[y[x],x])^2+2*(1-x)*D[y[x],x]-2*(x-y[x])==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [2 \text {arctanh}\left (\frac {(x-2) \sqrt {x^2-2 y(x)+1}-x^2+2 y(x)+2 x-1}{x \sqrt {x^2-2 y(x)+1}-x^2+2 y(x)-1}\right )-\sqrt {x^2-2 y(x)+1}+x&=c_1,y(x)\right ] \\ \text {Solve}\left [2 \text {arctanh}\left (\frac {x \sqrt {x^2-2 y(x)+1}-x^2+2 y(x)-1}{(x+2) \sqrt {x^2-2 y(x)+1}-x^2+2 y(x)-2 x-1}\right )+\sqrt {x^2-2 y(x)+1}+x&=c_1,y(x)\right ] \\ \end{align*}

✓ Sympy. Time used: 3.234 (sec). Leaf size: 51

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

\[ \left [ y{\left (x \right )} = \frac {x^{2}}{2} - \frac {\left (W\left (C_{1} e^{- x - 1}\right ) + 1\right )^{2}}{2} + \frac {1}{2}, \ y{\left (x \right )} = \frac {x^{2}}{2} - \frac {\left (W\left (C_{1} e^{- x - 1}\right ) + 1\right )^{2}}{2} + \frac {1}{2}\right ] \]