23.2.12 problem 12

Internal problem ID [5367]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 12:36:16 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 92

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

\begin{align*} y &= \frac {\sqrt {-b a}}{b} \\ y &= -\frac {\sqrt {-b a}}{b} \\ y &= \frac {-{\mathrm e}^{\sqrt {b}\, \left (c_1 -x \right )} a +{\mathrm e}^{\sqrt {b}\, \left (-c_1 +x \right )}}{2 \sqrt {b}} \\ y &= \frac {{\mathrm e}^{\sqrt {b}\, \left (c_1 -x \right )}-{\mathrm e}^{\sqrt {b}\, \left (-c_1 +x \right )} a}{2 \sqrt {b}} \\ \end{align*}

✓ Mathematica. Time used: 55.073 (sec). Leaf size: 123

ode=(D[y[x],x])^2==a+b*y[x]^2; 
ic={}; 
DSolve[{ode,ic},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 {i \sqrt {a}}{\sqrt {b}}\\ y(x)&\to \frac {i \sqrt {a}}{\sqrt {b}} \end{align*}

✓ Sympy. Time used: 10.927 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a - b*y(x)**2 + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\sqrt {a} \sinh {\left (\sqrt {b} \left (C_{1} - x\right ) \right )}}{\sqrt {b}}, \ y{\left (x \right )} = \frac {\sqrt {a} \sinh {\left (\sqrt {b} \left (C_{1} + x\right ) \right )}}{\sqrt {b}}\right ] \]