29.32.12 problem 946

Internal problem ID [5524]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 32
Problem number : 946
Date solved : Monday, January 27, 2025 at 11:35:19 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.181 (sec). Leaf size: 108

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

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {-4 \textit {\_a}^{2} b +a^{2}}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a +b \right )}d \textit {\_a} +2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {2 \textit {\_a}^{2}+a -\sqrt {-4 \textit {\_a}^{2} b +a^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+a +b \right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.789 (sec). Leaf size: 162

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

\begin{align*} \text {Solve}\left [\frac {a \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a\right )+(a+2 b) \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}-a-2 b\right )}{4 (a+b)}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {a \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}-a\right )+(a+2 b) \log \left (\sqrt {a^2-\frac {4 b y(x)^2}{x^2}}+a+2 b\right )}{4 (a+b)}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}