62.34.1 problem Ex 1

Internal problem ID [12993]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 58. Independent variable absent. Page 135
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:46:33 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}-y^{2} y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 27

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

\begin{align*} y &= 0 \\ y &= -\frac {c_{1} {\mathrm e}^{\left (x +c_{2} \right ) c_{1}}}{-1+{\mathrm e}^{\left (x +c_{2} \right ) c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.302 (sec). Leaf size: 93

DSolve[y[x]*D[y[x],{x,2}]-D[y[x],x]^2-y[x]^2*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+c_1 K[1]}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2-c_1 K[1]}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+c_1 K[1]}dK[1]\&\right ][x+c_2] \\ \end{align*}