29.27.18 problem 784

Internal problem ID [5368]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 27
Problem number : 784
Date solved : Sunday, March 30, 2025 at 08:03:44 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} {y^{\prime }}^{2}+x y^{\prime }+x -y&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 36

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

\[ y = -\frac {x^{2}}{4}+x +\operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{\frac {x}{2}-1}}{2}\right )^{2}+2 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{\frac {x}{2}-1}}{2}\right )+1 \]

✓ Mathematica. Time used: 3.053 (sec). Leaf size: 177

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

\begin{align*} \text {Solve}\left [-\sqrt {x^2+4 y(x)-4 x}+2 \log \left (\sqrt {x^2+4 y(x)-4 x}-x+2\right )-2 \log \left (-x \sqrt {x^2+4 y(x)-4 x}+x^2+4 y(x)-2 x-4\right )+x&=c_1,y(x)\right ] \\ \text {Solve}\left [-4 \text {arctanh}\left (\frac {(x-5) \sqrt {x^2+4 y(x)-4 x}-x^2-4 y(x)+7 x-6}{(x-3) \sqrt {x^2+4 y(x)-4 x}-x^2-4 y(x)+5 x-2}\right )+\sqrt {x^2+4 y(x)-4 x}+x&=c_1,y(x)\right ] \\ \end{align*}

✓ Sympy. Time used: 3.958 (sec). Leaf size: 75

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

\[ \left [ C_{1} + x + 2 \sqrt {\frac {x^{2}}{4} - x + y{\left (x \right )}} - 2 \log {\left (\sqrt {\frac {x^{2}}{4} - x + y{\left (x \right )}} + 1 \right )} = 0, \ C_{1} + x - 2 \sqrt {\frac {x^{2}}{4} - x + y{\left (x \right )}} - 2 \log {\left (\sqrt {\frac {x^{2}}{4} - x + y{\left (x \right )}} - 1 \right )} = 0\right ] \]