Internal problem ID [3426]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 24
Problem number: 687.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {\left (x^{2}-x^{3}+3 x y^{2}+2 y^{3}\right ) y^{\prime }+2 x^{3}+3 x^{2} y+y^{2}-y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 662
dsolve((x^2-x^3+3*x*y(x)^2+2*y(x)^3)*diff(y(x),x)+2*x^3+3*x^2*y(x)+y(x)^2-y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}+\frac {c_{1}}{6}\right )}{\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {x +\frac {c_{1}}{2}}{\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 x +c_{1}}{\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {x +\frac {c_{1}}{2}}{\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 x +c_{1}}{\left (-108 x^{3}-54 c_{1} x +6 \sqrt {324 x^{6}+324 x^{4} c_{1}+81 x^{2} c_{1}^{2}+48 x^{3}+72 c_{1} x^{2}+36 x c_{1}^{2}+6 c_{1}^{3}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 7.387 (sec). Leaf size: 348
DSolve[(x^2-x^3+3 x y[x]^2+2 y[x]^3)y'[x]+2 x^3+3 x^2 y[x]+y[x]^2-y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} (x+c_1)}{\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}-\frac {\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}{3 \sqrt [3]{2}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}{6 \sqrt [3]{2}}-\frac {\sqrt [3]{-2} (x+c_1)}{\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}} \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{2} (x+c_1)}{\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}{6 \sqrt [3]{2}} \\ y(x)\to -x \\ \end{align*}