40.14.10 problem 31

Internal problem ID [6781]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 19. Linear equations with variable coefficients (Misc. types). Supplemetary problems. Page 132
Problem number : 31
Date solved : Monday, January 27, 2025 at 02:30:20 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}&=y^{2} \ln \left (y\right ) \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 18

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

\[ y = {\mathrm e}^{\frac {c_1 \,{\mathrm e}^{-x}}{2}-\frac {{\mathrm e}^{x} c_2}{2}} \]

✓ Solution by Mathematica

Time used: 1.005 (sec). Leaf size: 63

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

\begin{align*} y(x)\to e^{\frac {1}{2} \left (e^{x+c_2}-c_1 e^{-x-c_2}\right )} \\ y(x)\to e^{\frac {1}{2} \left (e^{-x-c_2}-c_1 e^{x+c_2}\right )} \\ \end{align*}