83.23.23 problem 23

Internal problem ID [19304]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 23
Date solved : Tuesday, January 28, 2025 at 01:25:43 PM
CAS classification : [_rational]

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

✗ Solution by Maple

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

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 1.963 (sec). Leaf size: 123

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

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