82.40.7 problem Ex. 7

Internal problem ID [18953]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 94
Problem number : Ex. 7
Date solved : Tuesday, January 28, 2025 at 12:38:19 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2} y y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{2}-3 y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 55

dsolve(x^2*y(x)*diff(y(x),x$2)+(x*diff(y(x),x)-y(x))^2-3*y(x)^2=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\sqrt {10}\, \sqrt {-x \left (c_{1} x^{5}-c_{2} \right )}}{5 x} \\ y \left (x \right ) &= \frac {\sqrt {10}\, \sqrt {-x \left (c_{1} x^{5}-c_{2} \right )}}{5 x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.302 (sec). Leaf size: 23

DSolve[x^2*y[x]*D[y[x],{x,2}]+(x*D[y[x],x]-y[x])^2-3*y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {c_2 \sqrt {x^5+c_1}}{\sqrt {x}} \]