62.37.1 problem Ex 1

Internal problem ID [13008]
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 1
Date solved : Tuesday, January 28, 2025 at 08:24:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2} y y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 44

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

\begin{align*} y &= 0 \\ y &= \sqrt {2}\, \sqrt {-x \left (c_{1} x -c_{2} \right )} \\ y &= -\sqrt {2}\, \sqrt {-x \left (c_{1} x -c_{2} \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.257 (sec). Leaf size: 23

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

\[ y(x)\to c_2 \sqrt {x} \sqrt {2 x+c_1} \]