Internal problem ID [2335]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 37, page 171
Problem number: 22.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {\left ({y^{\prime }}^{2}+1\right ) x -\left (y+x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 36
dsolve((diff(y(x),x)^2+1)*x=diff(y(x),x)*(x+y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) = x y \left (x \right ) = \frac {x \left (\operatorname {LambertW}\left (\frac {x}{c_{1}}\right )^{2}-\operatorname {LambertW}\left (\frac {x}{c_{1}}\right )+1\right )}{\operatorname {LambertW}\left (\frac {x}{c_{1}}\right )} \end{align*}
✓ Solution by Mathematica
Time used: 3.512 (sec). Leaf size: 162
DSolve[(y'[x]^2+1)*x==y'[x]*(x+y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {1}{4} \left (-\frac {y(x)}{x}+\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+3}-4 \log \left (\sqrt {\frac {y(x)}{x}-1}-\sqrt {\frac {y(x)}{x}+3}\right )-3\right )=-\frac {\log (x)}{2}+c_1,y(x)\right ] \text {Solve}\left [\frac {1}{4} \left (\frac {y(x)}{x}+\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+3}-4 \log \left (\sqrt {\frac {y(x)}{x}-1}-\sqrt {\frac {y(x)}{x}+3}\right )+3\right )=\frac {\log (x)}{2}+c_1,y(x)\right ] \end{align*}