66.2.21 problem Problem 30

Internal problem ID [13920]
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
Date solved : Tuesday, January 28, 2025 at 06:08:37 AM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{2}&=\frac {y y^{\prime }}{\sqrt {x^{2}+1}} \end{align*}

✓ Solution by Maple

Time used: 0.070 (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 &= 0 \\ y &= \sqrt {c_{1} x \sqrt {x^{2}+1}+c_{1} x^{2}+c_{1} \operatorname {arcsinh}\left (x \right )+2 c_{2}} \\ y &= -\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: 66.228 (sec). Leaf size: 47

DSolve[y[x]*D[y[x],{x,2}]+D[y[x],x]^2== y[x]*D[y[x],x]/Sqrt[1+x^2],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\frac {e^{\text {arcsinh}(K[1])}}{\text {arcsinh}(K[1])+c_1+K[1] \left (K[1]+\sqrt {K[1]^2+1}\right )}dK[1]\right ) \]