2.60   ODE No. 60

\[ y'(x)-\frac {\sqrt {y(x)^2-1}}{\sqrt {x^2-1}}=0 \]

✓ Mathematica : cpu = 0.140767 (sec), leaf count = 92

DSolve[-(Sqrt[-1 + y[x]^2]/Sqrt[-1 + x^2]) + Derivative[1][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to -\frac {\tanh \left (\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right )+c_1\right )}{\sqrt {-1+\tanh ^2\left (\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right )+c_1\right )}}\right \},\left \{y(x)\to \frac {\tanh \left (\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right )+c_1\right )}{\sqrt {-1+\tanh ^2\left (\tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right )+c_1\right )}}\right \}\right \}\]

✓ Maple : cpu = 0.033 (sec), leaf count = 29

dsolve(diff(y(x),x)-(y(x)^2-1)^(1/2)/(x^2-1)^(1/2) = 0,y(x))
 

\[\ln \left (x +\sqrt {x^{2}-1}\right )-\ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-1}\right )+c_{1} = 0\]

Hand solution

\begin{equation} y^{\prime }=\pm \frac {\sqrt {y^{2}-1}}{\sqrt {x^{2}-1}}\tag {1}\end{equation}

Separable. For the positive case

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

Integrating

\[ \int \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\int \frac {dx}{\left ( x^{2}-1\right ) ^{\frac {1}{2}}}+C \]

But \(\int \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\tanh ^{-1}\frac {y}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\ln \left ( y+\sqrt {y^{2}-1}\right ) \), hence

\[ \ln \left ( y+\sqrt {y^{2}-1}\right ) =\ln \left ( x+\sqrt {x^{2}-1}\right ) +C \]

For the negative case

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

Integrating

\[ \int \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=-\int \frac {dx}{\left ( x^{2}-1\right ) ^{\frac {1}{2}}}+C \]

But \(\int \frac {dy}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\tanh ^{-1}\frac {y}{\left ( y^{2}-1\right ) ^{\frac {1}{2}}}=\ln \left ( y+\sqrt {y^{2}-1}\right ) \), hence

\[ \ln \left ( y+\sqrt {y^{2}-1}\right ) =-\ln \left ( x+\sqrt {x^{2}-1}\right ) +C \]

Therefore

\[ \ln \left ( y+\sqrt {y^{2}-1}\right ) =\pm \ln \left ( x+\sqrt {x^{2}-1}\right ) +C \]