29.32.10 problem 944

Internal problem ID [5522]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 32
Problem number : 944
Date solved : Monday, January 27, 2025 at 11:35:12 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.083 (sec). Leaf size: 113

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

\begin{align*} y \left (x \right ) &= x \sqrt {-a} \\ y \left (x \right ) &= -x \sqrt {-a} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {a \left (\textit {\_a}^{2}+a \right )}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a \right )}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-\sqrt {a \left (\textit {\_a}^{2}+a \right )}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a \right )}d \textit {\_a} +c_{1} \right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 9.674 (sec). Leaf size: 88

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

\begin{align*} y(x)\to -\sqrt {e^{c_1} \left (-2 \sqrt {a} x+e^{c_1}\right )} \\ y(x)\to \sqrt {e^{c_1} \left (-2 \sqrt {a} x+e^{c_1}\right )} \\ y(x)\to 0 \\ y(x)\to -i \sqrt {a} x \\ y(x)\to i \sqrt {a} x \\ \end{align*}