54.1.58 problem 59

Internal problem ID [11372]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 59
Date solved : Tuesday, September 30, 2025 at 08:06:21 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }-a \sqrt {y^{2}+1}-b&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

ode:=diff(y(x),x)-a*(1+y(x)^2)^(1/2)-b = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ x -\int _{}^{y}\frac {1}{a \sqrt {\textit {\_a}^{2}+1}+b}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.256 (sec). Leaf size: 78

ode=D[y[x],x] - a*Sqrt[y[x]^2+1] - b==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\sqrt {K[1]^2+1} a+b}dK[1]\&\right ][x+c_1]\\ y(x)&\to -\frac {\sqrt {b^2-a^2}}{a}\\ y(x)&\to \frac {\sqrt {b^2-a^2}}{a} \end{align*}

✓ Sympy. Time used: 0.602 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a*sqrt(y(x)**2 + 1) - b + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \int \limits ^{y{\left (x \right )}} \frac {1}{a \sqrt {y^{2} + 1} + b}\, dy = C_{1} - x \]