Internal problem ID [1011]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable
Equations. Section 2.4 Page 68
Problem number: 34.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x^{3}+y x^{2}+3 y^{3}}{x^{3}+3 x y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 369
dsolve(diff(y(x),x)=(x^3+x^2*y(x)+3*y(x)^3)/(x^3+3*x*y(x)^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (\frac {\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right ) x \\ y \relax (x ) = \left (-\frac {\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ y \relax (x ) = \left (-\frac {\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \ln \relax (x )+108 c_{1}+12 \sqrt {12+81 \ln \relax (x )^{2}+162 \ln \relax (x ) c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.297 (sec). Leaf size: 350
DSolve[y'[x]==(x^3+x^2*y[x]+3*y[x]^3)/(x^3+3*x*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 \sqrt [3]{3} x^2+\sqrt [3]{2} \left (\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 (\log (x)+c_1)\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 (\log (x)+c_1)}} \\ y(x)\to \frac {4 \sqrt [3]{-3} x^2+i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (\sqrt {3} \sqrt {x^6 \left (27 \log (x) (\log (x)+2 c_1)+4+27 c_1{}^2\right )}+9 x^3 (\log (x)+c_1)\right ){}^{2/3}}{2\ 6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 (\log (x)+c_1)}} \\ y(x)\to \frac {x^2 \text {Root}\left [\text {$\#$1}^3+24\&,2\right ]-\sqrt [3]{-2} \left (\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 (\log (x)+c_1)\right ){}^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {x^6 \left (4+27 (\log (x)+c_1){}^2\right )}+9 x^3 (\log (x)+c_1)}} \\ \end{align*}