Internal problem ID [3366]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 22
Problem number: 626.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {\left (x^{2}-3 y^{2}\right ) y^{\prime }+1+2 y x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 392
dsolve((x^2-3*y(x)^2)*diff(y(x),x)+1+2*x*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{2}}{\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2 x^{2}}{\left (108 x +108 c_{1}+12 \sqrt {-12 x^{6}+81 c_{1}^{2}+162 c_{1} x +81 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.796 (sec). Leaf size: 310
DSolve[(x^2-3 y[x]^2)y'[x]+1+2 x y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{6 \sqrt [3]{2}}+\frac {\sqrt [3]{-\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {-4 x^6+27 x^2-54 c_1 x+27 c_1{}^2}-9 x+9 c_1}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}}{6 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {-108 x^6+729 (x-c_1){}^2}-27 x+27 c_1}} \\ \end{align*}