29.26.23 problem 759

Internal problem ID [5344]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 759
Date solved : Monday, January 27, 2025 at 11:15:58 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 99

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

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

✓ Solution by Mathematica

Time used: 1.225 (sec). Leaf size: 92

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

\begin{align*} y(x)\to -\frac {\sqrt {a} \sinh \left (\sqrt {b} (x-c_1)\right )}{\sqrt {b}} \\ y(x)\to \frac {\sqrt {a} \sinh \left (\sqrt {b} (x+c_1)\right )}{\sqrt {b}} \\ y(x)\to -\frac {i \sqrt {a}}{\sqrt {b}} \\ y(x)\to \frac {i \sqrt {a}}{\sqrt {b}} \\ \end{align*}