56.1.80 problem 79

Internal problem ID [8792]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 79
Date solved : Sunday, March 30, 2025 at 01:36:02 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {y \left (1+\frac {a^{2} x}{\sqrt {a^{2} \left (x^{2}+1\right )}}\right )}{\sqrt {a^{2} \left (x^{2}+1\right )}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

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

\[ y = c_1 \left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a^{2} \left (x^{2}+1\right )}\right )^{\frac {1}{\sqrt {a^{2}}}} \sqrt {x^{2}+1} \]

✓ Mathematica. Time used: 0.298 (sec). Leaf size: 116

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

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

✓ Sympy. Time used: 15.841 (sec). Leaf size: 56

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

\[ y{\left (x \right )} = C_{1} \sqrt {x^{2} + 1} e^{\int \frac {1}{\sqrt {a^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx + \int \frac {x^{2}}{\sqrt {a^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx} \]