29.25.31 problem 728

Internal problem ID [5318]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 728
Date solved : Tuesday, March 04, 2025 at 09:22:02 PM
CAS classification : [_separable]

\begin{align*} \left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{{3}/{2}} y^{\prime }&=1+y^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

ode:=(y(x)+(1+y(x)^2)^(1/2))*(x^2+1)^(3/2)*diff(y(x),x) = 1+y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {x}{\sqrt {x^{2}+1}}-\operatorname {arcsinh}\left (y \left (x \right )\right )-\frac {\ln \left (y \left (x \right )^{2}+1\right )}{2}+c_{1} = 0 \]

✓ Mathematica. Time used: 0.513 (sec). Leaf size: 115

ode=(1+x^2)^(3/2)*(y[x]+Sqrt[1+y[x]^2])*D[y[x],x]==1+y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.252 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 1)**(3/2)*(sqrt(y(x)**2 + 1) + y(x))*Derivative(y(x), x) - y(x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \frac {x}{\sqrt {x^{2} + 1}} + \frac {\log {\left (y^{2}{\left (x \right )} + 1 \right )}}{2} + \operatorname {asinh}{\left (y{\left (x \right )} \right )} = C_{1} \]