Internal problem ID [3393]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 654.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {\left (1-4 x +3 x y^{2}\right ) y^{\prime }-\left (2-y^{2}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 24
dsolve((1-4*x+3*x*y(x)^2)*diff(y(x),x) = (2-y(x)^2)*y(x),y(x), singsol=all)
\[ x +\frac {1}{y \relax (x )^{2}}-\frac {c_{1}}{\sqrt {y \relax (x )^{2}-2}\, y \relax (x )^{2}} = 0 \]
✓ Solution by Mathematica
Time used: 62.843 (sec). Leaf size: 1937
DSolve[(1-4 x+3 x y[x]^2)y'[x]==(2-y[x]^2)y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\frac {8 \sqrt [3]{2} x^4+8 \sqrt [3]{2} x^3-4 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+\left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}+2 x^2 \left (\sqrt [3]{2}+2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {\frac {8 \sqrt [3]{2} x^4+8 \sqrt [3]{2} x^3-4 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+\left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}+2 x^2 \left (\sqrt [3]{2}+2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{\sqrt {6}} \\ y(x)\to -\frac {\sqrt {\frac {4 x^2 \left (\text {Root}\left [\text {$\#$1}^3-2\&,3\right ]+2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )+16 (-1)^{2/3} \sqrt [3]{2} x^4+16 (-1)^{2/3} \sqrt [3]{2} x^3-8 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+\left (-1-i \sqrt {3}\right ) \left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{2 \sqrt {3}} \\ y(x)\to \frac {\sqrt {\frac {4 x^2 \left (\text {Root}\left [\text {$\#$1}^3-2\&,3\right ]+2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )+16 (-1)^{2/3} \sqrt [3]{2} x^4+16 (-1)^{2/3} \sqrt [3]{2} x^3-8 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+\left (-1-i \sqrt {3}\right ) \left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{2 \sqrt {3}} \\ y(x)\to -\frac {\sqrt {\frac {-16 \sqrt [3]{-2} x^4-16 \sqrt [3]{-2} x^3-8 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+i \left (\sqrt {3}+i\right ) \left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}-4 x^2 \left (\sqrt [3]{-2}-2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{2 \sqrt {3}} \\ y(x)\to \frac {\sqrt {\frac {-16 \sqrt [3]{-2} x^4-16 \sqrt [3]{-2} x^3-8 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+i \left (\sqrt {3}+i\right ) \left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}-4 x^2 \left (\sqrt [3]{-2}-2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{2 \sqrt {3}} \\ y(x)\to 0 \\ y(x)\to -\sqrt {2} \\ y(x)\to \sqrt {2} \\ \end{align*}