2.60   ODE No. 60

  1. Problem in Latex
  2. Mathematica input
  3. Maple input

\[ y'(x)-\frac {\sqrt {y(x)^2-1}}{\sqrt {x^2-1}}=0 \] Mathematica : cpu = 0.0494737 (sec), leaf count = 55

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

Maple : cpu = 0.016 (sec), leaf count = 29

\[ \left \{ \ln \left ( x+\sqrt {{x}^{2}-1} \right ) -\ln \left ( y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}-1} \right ) +{\it \_C1}=0 \right \} \]

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

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