23.1.107 problem 105

Internal problem ID [4714]
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 : 105
Date solved : Tuesday, September 30, 2025 at 08:18:35 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 9.132 (sec). Leaf size: 42

ode=D[y[x],x]==y[x]*Sqrt[a+b*y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {a \text {sech}^2\left (\frac {1}{2} \sqrt {a} (x+c_1)\right )}{b}\\ y(x)&\to 0\\ y(x)&\to -\frac {a}{b} \end{align*}

✓ Sympy. Time used: 0.764 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \frac {2 a}{b \left (\cosh {\left (\sqrt {a} \left (C_{1} + x\right ) \right )} - 1\right )} \]