20.4.25 problem Problem 41

Internal problem ID [3660]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 41
Date solved : Sunday, March 30, 2025 at 02:02:31 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }+\frac {2 y}{x}&=6 \sqrt {x^{2}+1}\, \sqrt {y} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 38

ode:=diff(y(x),x)+2*y(x)/x = 6*(x^2+1)^(1/2)*y(x)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {-\sqrt {x^{2}+1}\, x^{2}+\sqrt {y}\, x -c_1 -\sqrt {x^{2}+1}}{x} = 0 \]

✓ Mathematica. Time used: 0.24 (sec). Leaf size: 55

ode=D[y[x],x]+2/x*y[x]==6*Sqrt[1+x^2]*Sqrt[y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {x^6+3 x^4+x^2 \left (3+2 c_1 \sqrt {x^2+1}\right )+2 c_1 \sqrt {x^2+1}+1+c_1{}^2}{x^2} \]

✓ Sympy. Time used: 0.340 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*sqrt(x**2 + 1)*sqrt(y(x)) + Derivative(y(x), x) + 2*y(x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1}^{2}}{x^{2}} + 2 C_{1} \sqrt {x^{2} + 1} + \frac {2 C_{1} \sqrt {x^{2} + 1}}{x^{2}} + x^{4} + 3 x^{2} + 3 + \frac {1}{x^{2}} \]