83.23.26 problem 26

Internal problem ID [19307]
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 : 26
Date solved : Tuesday, January 28, 2025 at 01:28:47 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.325 (sec). Leaf size: 854

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

\begin{align*} y \left (x \right ) &= 1-x \\ y \left (x \right ) &= -x -1 \\ y \left (x \right ) &= x -1 \\ y \left (x \right ) &= x +1 \\ y \left (x \right ) &= 0 \\ \int _{\textit {\_b}}^{x}\frac {y \left (x \right )^{2}+\textit {\_a}^{2}-1+\sqrt {y \left (x \right )^{4}+\left (-2 \textit {\_a}^{2}-2\right ) y \left (x \right )^{2}+\textit {\_a}^{4}-2 \textit {\_a}^{2}+1}}{\textit {\_a} \left (\left (-y \left (x \right )^{2}+\textit {\_a}^{2}-1\right ) \sqrt {y \left (x \right )^{4}+\left (-2 \textit {\_a}^{2}-2\right ) y \left (x \right )^{2}+\textit {\_a}^{4}-2 \textit {\_a}^{2}+1}+y \left (x \right )^{4}+\left (-2 \textit {\_a}^{2}-2\right ) y \left (x \right )^{2}+\textit {\_a}^{4}-2 \textit {\_a}^{2}+1\right )}d \textit {\_a} -2 \left (\int _{}^{y \left (x \right )}\frac {4 \left (\frac {1}{4}+\left (\left (-\textit {\_f}^{2}+x^{2}-1\right ) \sqrt {\textit {\_f}^{4}+\left (-2 x^{2}-2\right ) \textit {\_f}^{2}+x^{4}-2 x^{2}+1}+\textit {\_f}^{4}+\left (-2 x^{2}-2\right ) \textit {\_f}^{2}+x^{4}-2 x^{2}+1\right ) \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a} \left (\left (\textit {\_a}^{2}-\textit {\_f}^{2}-1\right ) \sqrt {\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}-2 \textit {\_f}^{2}+1}+\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}+1\right )}{\sqrt {\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}-2 \textit {\_f}^{2}+1}\, {\left (\left (\textit {\_a}^{2}-\textit {\_f}^{2}-1\right ) \sqrt {\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}-2 \textit {\_f}^{2}+1}+\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}-2 \textit {\_f}^{2}+1\right )}^{2}}d \textit {\_a} \right )\right ) \textit {\_f}}{\left (-\textit {\_f}^{2}+x^{2}-1\right ) \sqrt {\textit {\_f}^{4}+\left (-2 x^{2}-2\right ) \textit {\_f}^{2}+x^{4}-2 x^{2}+1}+\textit {\_f}^{4}+\left (-2 x^{2}-2\right ) \textit {\_f}^{2}+x^{4}-2 x^{2}+1}d \textit {\_f} \right )+c_{1} &= 0 \\ \int _{\textit {\_b}}^{x}\frac {y \left (x \right )^{2}+\textit {\_a}^{2}-\sqrt {y \left (x \right )^{4}+\left (-2 \textit {\_a}^{2}-2\right ) y \left (x \right )^{2}+\textit {\_a}^{4}-2 \textit {\_a}^{2}+1}-1}{\left (\left (y \left (x \right )^{2}-\textit {\_a}^{2}+1\right ) \sqrt {y \left (x \right )^{4}+\left (-2 \textit {\_a}^{2}-2\right ) y \left (x \right )^{2}+\textit {\_a}^{4}-2 \textit {\_a}^{2}+1}+y \left (x \right )^{4}+\left (-2 \textit {\_a}^{2}-2\right ) y \left (x \right )^{2}+\textit {\_a}^{4}-2 \textit {\_a}^{2}+1\right ) \textit {\_a}}d \textit {\_a} -2 \left (\int _{}^{y \left (x \right )}\frac {4 \textit {\_f} \left (\frac {1}{4}+\left (\left (\textit {\_f}^{2}-x^{2}+1\right ) \sqrt {\textit {\_f}^{4}+\left (-2 x^{2}-2\right ) \textit {\_f}^{2}+x^{4}-2 x^{2}+1}+\textit {\_f}^{4}+\left (-2 x^{2}-2\right ) \textit {\_f}^{2}+x^{4}-2 x^{2}+1\right ) \left (\int _{\textit {\_b}}^{x}-\frac {\left (\left (-\textit {\_a}^{2}+\textit {\_f}^{2}+1\right ) \sqrt {\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}-2 \textit {\_f}^{2}+1}+\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}+1\right ) \textit {\_a}}{\sqrt {\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}-2 \textit {\_f}^{2}+1}\, {\left (\left (-\textit {\_a}^{2}+\textit {\_f}^{2}+1\right ) \sqrt {\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}-2 \textit {\_f}^{2}+1}+\textit {\_a}^{4}+\left (-2 \textit {\_f}^{2}-2\right ) \textit {\_a}^{2}+\textit {\_f}^{4}-2 \textit {\_f}^{2}+1\right )}^{2}}d \textit {\_a} \right )\right )}{\left (\textit {\_f}^{2}-x^{2}+1\right ) \sqrt {\textit {\_f}^{4}+\left (-2 x^{2}-2\right ) \textit {\_f}^{2}+x^{4}-2 x^{2}+1}+\textit {\_f}^{4}+\left (-2 x^{2}-2\right ) \textit {\_f}^{2}+x^{4}-2 x^{2}+1}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.276 (sec). Leaf size: 54

DSolve[x*y[x]*D[y[x],x]^2-(x^2+y[x]^2-1)*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 {1}{-1+c_1}\right )} \\ y(x)\to -x-1 \\ y(x)\to 1-x \\ y(x)\to x-1 \\ y(x)\to x+1 \\ \end{align*}