29.14.22 problem 403

Internal problem ID [5001]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 403
Date solved : Tuesday, March 04, 2025 at 07:42:07 PM
CAS classification : [_separable]

\begin{align*} y^{\prime } \sqrt {x^{4}+x^{2}+1}&=\sqrt {1+y^{2}+y^{4}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

ode:=diff(y(x),x)*(x^4+x^2+1)^(1/2) = (1+y(x)^2+y(x)^4)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \int \frac {1}{\sqrt {x^{4}+x^{2}+1}}d x -\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{4}+\textit {\_a}^{2}+1}}d \textit {\_a} +c_{1} = 0 \]

✓ Mathematica. Time used: 41.715 (sec). Leaf size: 189

ode=D[y[x],x]Sqrt[1+x^2+x^4]==Sqrt[1+y[x]^2+y[x]^4]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {(-1)^{2/3} \sqrt {\sqrt [3]{-1} \text {$\#$1}^2+1} \sqrt {1-(-1)^{2/3} \text {$\#$1}^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} \text {$\#$1}\right ),(-1)^{2/3}\right )}{\sqrt {\text {$\#$1}^4+\text {$\#$1}^2+1}}\&\right ]\left [\frac {(-1)^{2/3} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )}{\sqrt {x^4+x^2+1}}+c_1\right ] \\ y(x)\to -\sqrt [3]{-1} \\ y(x)\to \sqrt [3]{-1} \\ y(x)\to -(-1)^{2/3} \\ y(x)\to (-1)^{2/3} \\ \end{align*}

✓ Sympy. Time used: 0.661 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sqrt(x**4 + x**2 + 1)*Derivative(y(x), x) - sqrt(y(x)**4 + y(x)**2 + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {y^{4} + y^{2} + 1}}\, dy = C_{1} + \int \frac {1}{\sqrt {x^{4} + x^{2} + 1}}\, dx \]