75.14.34 problem 360

Internal problem ID [16948]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 360
Date solved : Tuesday, January 28, 2025 at 09:43:05 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 } \end{align*}

✓ Solution by Maple

Time used: 0.219 (sec). Leaf size: 27

dsolve(y(x)*diff(y(x),x$2)-diff(y(x),x)^2=y(x)^2*diff(y(x),x),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: 0.337 (sec). Leaf size: 93

DSolve[y[x]*D[y[x],{x,2}]-D[y[x],x]^2==y[x]^2*D[y[x],x],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*}