Internal problem ID [3657]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 403.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime } \sqrt {x^{4}+x^{2}+1}-\sqrt {1+y^{2}+y^{4}}=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ \int \frac {1}{\sqrt {x^{4}+x^{2}+1}}d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{4}+\textit {\_a}^{2}+1}}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 41.523 (sec). Leaf size: 189
DSolve[y'[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*}