82.15.3 problem Ex. 3

Internal problem ID [18813]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 35
Problem number : Ex. 3
Date solved : Tuesday, January 28, 2025 at 12:21:19 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 91

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

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

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 99

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

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