81.7.10 problem 11

Internal problem ID [18628]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter VI. Certain particular forms of equations. Exercises at page 74
Problem number : 11
Date solved : Thursday, March 13, 2025 at 12:26:52 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 31

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

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

✓ Mathematica. Time used: 14.855 (sec). Leaf size: 98

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

\begin{align*} y(x)\to -W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^2-2 W\left (e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right )+x-1 \\ y(x)\to -W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right ){}^2-2 W\left (-e^{\frac {1}{2} (-x-2+c_1)}\right )+x-1 \\ y(x)\to x-1 \\ \end{align*}

✓ Sympy. Time used: 1.757 (sec). Leaf size: 54

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

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