83.22.10 problem 10

Internal problem ID [19264]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 01:17:37 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.183 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.531 (sec). Leaf size: 161

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

\begin{align*} y(x)\to -x \cos \left (2 \arctan \left (\frac {\left (\sqrt {x-1}-1\right ) \left (\sqrt {3}-\sqrt {x+1}\right )}{\left (\sqrt {x-1}+1\right ) \left (\sqrt {3} \sqrt {x+1}-1\right )-2 x}\right )+i c_1\right ) \\ y(x)\to -x \cos \left (2 \arctan \left (\frac {\left (\sqrt {x-1}-1\right ) \left (\sqrt {3}-\sqrt {x+1}\right )}{\left (\sqrt {x-1}+1\right ) \left (\sqrt {3} \sqrt {x+1}-1\right )-2 x}\right )-i c_1\right ) \\ y(x)\to -x \\ y(x)\to x \\ \end{align*}