81.3.14 problem 14

Internal problem ID [18632]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 12:04:55 PM
CAS classification : [_separable]

\begin{align*} y \sqrt {x^{2}-1}+x \sqrt {-1+y^{2}}\, y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 0.461 (sec). Leaf size: 60

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\sqrt {\text {$\#$1}^2-1}-\arctan \left (\sqrt {\text {$\#$1}^2-1}\right )\&\right ]\left [\arctan \left (\sqrt {x^2-1}\right )-\sqrt {x^2-1}+c_1\right ] \\ y(x)\to 0 \\ \end{align*}