23.1.106 problem 104

Internal problem ID [4713]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 104
Date solved : Tuesday, September 30, 2025 at 08:18:32 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 35

ode:=diff(y(x),x) = (a+b*y(x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\left (c_1 +x \right ) \sqrt {b}-\ln \left (\sqrt {b}\, y+\sqrt {a +b y^{2}}\right )}{\sqrt {b}} = 0 \]

✓ Mathematica. Time used: 39.056 (sec). Leaf size: 78

ode=D[y[x],x]==Sqrt[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 {i \sqrt {a}}{\sqrt {b}}\\ y(x)&\to \frac {i \sqrt {a}}{\sqrt {b}} \end{align*}

✓ Sympy. Time used: 0.480 (sec). Leaf size: 22

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

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