29.34.5 problem 1000

Internal problem ID [5576]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 1000
Date solved : Monday, January 27, 2025 at 12:10:41 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }+a^{2} x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.149 (sec). Leaf size: 152

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

\begin{align*} y \left (x \right ) &= \sqrt {2}\, \sqrt {-a x} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {-a x} \\ y \left (x \right ) &= \sqrt {2}\, \sqrt {a x} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {a x} \\ y \left (x \right ) &= \frac {{\mathrm e}^{\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (16 x \,a^{2} {\mathrm e}^{2 \textit {\_Z} +2 c_{1}}+{\mathrm e}^{2 \textit {\_Z}} x^{3}-4 \,{\mathrm e}^{3 \textit {\_Z} +2 c_{1}}\right )}{2}}}{\sqrt {x}} \\ y \left (x \right ) &= \sqrt {x}\, {\mathrm e}^{-\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (x^{2} \left (16 a^{2} x^{2} {\mathrm e}^{2 \textit {\_Z} -2 c_{1}}-4 \,{\mathrm e}^{-2 c_{1} +3 \textit {\_Z}} x +{\mathrm e}^{2 \textit {\_Z}}\right )\right )}{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 18.764 (sec). Leaf size: 219

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

\begin{align*} y(x)\to -\sqrt {-2 a^2 e^{-c_1} x^2-\frac {e^{c_1}}{2}} \\ y(x)\to \sqrt {-2 a^2 e^{-c_1} x^2-\frac {e^{c_1}}{2}} \\ y(x)\to -\frac {\sqrt {4 a^2 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {4 a^2 e^{-c_1} x^2+e^{c_1}}}{\sqrt {2}} \\ y(x)\to -\sqrt {2} \sqrt {a} \sqrt {x} \\ y(x)\to -i \sqrt {2} \sqrt {a} \sqrt {x} \\ y(x)\to i \sqrt {2} \sqrt {a} \sqrt {x} \\ y(x)\to \sqrt {2} \sqrt {a} \sqrt {x} \\ \end{align*}