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 : Monday, January 27, 2025 at 10:02:34 AM
CAS classification : [_separable]

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

✓ Solution by Maple

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

dsolve(diff(y(x),x)*sqrt(x^4+x^2+1) = sqrt(1+y(x)^2+y(x)^4),y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]Sqrt[1+x^2+x^4]==Sqrt[1+y[x]^2+y[x]^4],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*}