60.1.445 problem 448

Internal problem ID [10459]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 448
Date solved : Monday, January 27, 2025 at 07:46:07 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.086 (sec). Leaf size: 162

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

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

✓ Solution by Mathematica

Time used: 0.267 (sec). Leaf size: 98

DSolve[1 - y[x]^2 + (-1 + x^2)*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

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