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 : Monday, January 27, 2025 at 11:17:32 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*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 32

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

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

✓ Solution by Mathematica

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

DSolve[(D[y[x],x])^2+2*(1-x)*D[y[x],x]-2*(x-y[x])==0,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*}