Internal problem ID [11864]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 30.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {{y^{\prime }}^{2}+y y^{\prime \prime }-\frac {y y^{\prime }}{\sqrt {x^{2}+1}}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 63
dsolve(y(x)*diff(y(x),x$2)+diff(y(x),x)^2= y(x)*diff(y(x),x)/sqrt(1+x^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) = 0 y \left (x \right ) = \sqrt {c_{1} x \sqrt {x^{2}+1}+c_{1} x^{2}+c_{1} \operatorname {arcsinh}\left (x \right )+2 c_{2}} y \left (x \right ) = -\sqrt {c_{1} x \sqrt {x^{2}+1}+c_{1} x^{2}+c_{1} \operatorname {arcsinh}\left (x \right )+2 c_{2}} \end{align*}
✓ Solution by Mathematica
Time used: 60.936 (sec). Leaf size: 73
DSolve[y[x]*y''[x]+y'[x]^2== y[x]*y'[x]/Sqrt[1+x^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \exp \left (\int _1^x\frac {1}{-K[1] c_1+\sqrt {K[1]^2+1} c_1+K[1]+\left (K[1]-\sqrt {K[1]^2+1}\right ) \log \left (\sqrt {K[1]^2+1}-K[1]\right )}dK[1]\right ) \]