29.26.21 problem 757

Internal problem ID [5342]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 757
Date solved : Monday, January 27, 2025 at 11:15:55 AM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}&=a^{2}-y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 60

dsolve(diff(y(x),x)^2 = a^2-y(x)^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 3.514 (sec). Leaf size: 111

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

\begin{align*} y(x)\to -\frac {a \tan (x-c_1)}{\sqrt {\sec ^2(x-c_1)}} \\ y(x)\to \frac {a \tan (x-c_1)}{\sqrt {\sec ^2(x-c_1)}} \\ y(x)\to -\frac {a \tan (x+c_1)}{\sqrt {\sec ^2(x+c_1)}} \\ y(x)\to \frac {a \tan (x+c_1)}{\sqrt {\sec ^2(x+c_1)}} \\ y(x)\to -a \\ y(x)\to a \\ \end{align*}