69.1.64 problem 91

Internal problem ID [14217]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 91
Date solved : Tuesday, January 28, 2025 at 06:22:22 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.078 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 3.077 (sec). Leaf size: 177

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