60.1.451 problem 454

Internal problem ID [10465]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 454
Date solved : Monday, January 27, 2025 at 07:49:06 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.123 (sec). Leaf size: 106

dsolve(a*x^2*diff(y(x),x)^2-2*a*x*y(x)*diff(y(x),x)+y(x)^2-a*(a-1)*x^2 = 0,y(x), singsol=all)
 

\begin{align*} y &= \sqrt {-a}\, x \\ y &= -\sqrt {-a}\, x \\ y &= \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 &= \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: 0.348 (sec). Leaf size: 113

DSolve[-((-1 + a)*a*x^2) + y[x]^2 - 2*a*x*y[x]*D[y[x],x] + a*x^2*D[y[x],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*}