29.31.25 problem 925

Internal problem ID [5504]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 925
Date solved : Monday, January 27, 2025 at 11:33:00 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} a \,x^{2} {y^{\prime }}^{2}-2 a x y y^{\prime }+a \left (1-a \right ) x^{2}+y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.144 (sec). Leaf size: 106

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

✓ Solution by Mathematica

Time used: 0.327 (sec). Leaf size: 113

DSolve[a x^2 (D[y[x],x])^2-2 a x y[x] D[y[x],x]+a(1-a)x^2+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{2} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-a x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right ) \\ y(x)\to \frac {1}{2} e^{c_1} x^{\sqrt {\frac {a-1}{a}}+1}-\frac {1}{2} a e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \\ \end{align*}