82.15.4 problem Ex. 4

Internal problem ID [18814]
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. 4
Date solved : Tuesday, January 28, 2025 at 12:21:20 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 73

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

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

✓ Solution by Mathematica

Time used: 0.353 (sec). Leaf size: 86

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

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