Internal problem ID [3492]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 26
Problem number: 759.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-a -b y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.096 (sec). Leaf size: 114
dsolve(diff(y(x),x)^2 = a+b*y(x)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {-a b}}{b} \\ y \relax (x ) = -\frac {\sqrt {-a b}}{b} \\ y \relax (x ) = \frac {\left ({\mathrm e}^{-2 c_{1} \sqrt {b}} {\mathrm e}^{2 x \sqrt {b}}-a \right ) {\mathrm e}^{c_{1} \sqrt {b}} {\mathrm e}^{-x \sqrt {b}}}{2 \sqrt {b}} \\ y \relax (x ) = \frac {\left ({\mathrm e}^{2 c_{1} \sqrt {b}} {\mathrm e}^{-2 x \sqrt {b}}-a \right ) {\mathrm e}^{-c_{1} \sqrt {b}} {\mathrm e}^{x \sqrt {b}}}{2 \sqrt {b}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 140.013 (sec). Leaf size: 207
DSolve[(y'[x])^2==a+b y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {a} \tanh \left (\sqrt {b} (x-c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x-c_1)\right )}} \\ y(x)\to \frac {\sqrt {a} \tanh \left (\sqrt {b} (x-c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x-c_1)\right )}} \\ y(x)\to -\frac {\sqrt {a} \tanh \left (\sqrt {b} (x+c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x+c_1)\right )}} \\ y(x)\to \frac {\sqrt {a} \tanh \left (\sqrt {b} (x+c_1)\right )}{\sqrt {b \text {sech}^2\left (\sqrt {b} (x+c_1)\right )}} \\ y(x)\to -\frac {i \sqrt {a}}{\sqrt {b}} \\ y(x)\to \frac {i \sqrt {a}}{\sqrt {b}} \\ \end{align*}