83.23.3 problem 3

Internal problem ID [19284]
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 : 3
Date solved : Tuesday, January 28, 2025 at 01:24:25 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 60

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

\begin{align*} y \left (x \right ) &= \sqrt {-m \left (x^{2}-1\right )} \\ y \left (x \right ) &= -\sqrt {\left (-x^{2}+1\right ) m} \\ y \left (x \right ) &= c_{1} x -\sqrt {c_{1}^{2}+m} \\ y \left (x \right ) &= c_{1} x +\sqrt {c_{1}^{2}+m} \\ \end{align*}

✓ Solution by Mathematica

Time used: 13.914 (sec). Leaf size: 101

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

\begin{align*} y(x)\to -\sqrt {m} \sec \left (c_1 \sqrt {m}\right ) \left (-1+x \sin \left (c_1 \sqrt {m}\right )\right ) \\ y(x)\to -\sqrt {m} \sec \left (c_1 \sqrt {m}\right ) \left (1+x \sin \left (c_1 \sqrt {m}\right )\right ) \\ y(x)\to -\sqrt {m-m x^2} \\ y(x)\to \sqrt {m-m x^2} \\ \end{align*}