81.7.10 problem 11

Internal problem ID [18707]
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 : Tuesday, January 28, 2025 at 12:12:07 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[x==y[x]+D[y[x],x]^2,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*}