62.37.3 problem Ex 3

Internal problem ID [13010]
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 61. Transformation of variables. Page 143
Problem number : Ex 3
Date solved : Tuesday, January 28, 2025 at 08:24:47 PM
CAS classification : [[_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.158 (sec). Leaf size: 22

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

\[ y = {\mathrm e}^{x^{2}+2-\frac {{\mathrm e}^{x} c_{2}}{2}+\frac {c_{1} {\mathrm e}^{-x}}{2}} \]

✓ Solution by Mathematica

Time used: 60.199 (sec). Leaf size: 275

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

\[ y(x)\to -\frac {\exp \left (-\int _1^x\frac {e^{-K[3]} \left (y(K[3]) \left (e^{2 K[3]} c_1{}^2-e^{K[3]} c_1-K[3]^2+e^{2 K[3]} \int _1^{K[3]}\frac {e^{-K[1]} \left (\left (K[1]^2-\log (y(K[1]))\right ) y(K[1])+y''(K[1])\right )}{y(K[1])}dK[1]{}^2+\log (y(K[3]))+e^{K[3]} \left (2 e^{K[3]} c_1-1\right ) \int _1^{K[3]}\frac {e^{-K[1]} \left (\left (K[1]^2-\log (y(K[1]))\right ) y(K[1])+y''(K[1])\right )}{y(K[1])}dK[1]\right )-y''(K[3])\right )}{y(K[3]) \left (c_1+\int _1^{K[3]}\frac {e^{-K[1]} \left (\left (K[1]^2-\log (y(K[1]))\right ) y(K[1])+y''(K[1])\right )}{y(K[1])}dK[1]\right )}dK[3]-x+c_2\right )}{\int _1^x\frac {e^{-K[1]} \left (\left (K[1]^2-\log (y(K[1]))\right ) y(K[1])+y''(K[1])\right )}{y(K[1])}dK[1]+c_1} \]