Internal problem ID [3781]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1070.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{4} \left (y^{\prime }\right )^{3}-x^{3} y \left (y^{\prime }\right )^{2}-x^{2} y^{2} y^{\prime }+x y^{3}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.672 (sec). Leaf size: 464
dsolve(x^4*diff(y(x),x)^3-x^3*y(x)*diff(y(x),x)^2-x^2*y(x)^2*diff(y(x),x)+x*y(x)^3 = 1,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4 x} \\ y \relax (x ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{8 x}-\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{8 x} \\ y \relax (x ) = -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{8 x}+\frac {3 i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{8 x} \\ y \relax (x ) = \frac {\RootOf \left (-\ln \relax (x )+3 \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 \,2^{\frac {2}{3}} \textit {\_a}^{2}+2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}+\left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {2}{3}}}d \textit {\_a} \right )+c_{1}\right )}{x^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}-\frac {3 \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}} \left (\sqrt {3}+i\right )}{2 \left (2 \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i 2^{\frac {2}{3}} \textit {\_a}^{2}-\textit {\_a} \sqrt {3}\, \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}-i \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}+i \left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {2}{3}}\right )}d \textit {\_a} +c_{1}\right )}{x^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}-\frac {3 \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}} \left (-i+\sqrt {3}\right )}{2 \left (2 \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}+2 i 2^{\frac {2}{3}} \textit {\_a}^{2}-\textit {\_a} \sqrt {3}\, \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}+i \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}-i \left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {2}{3}}\right )}d \textit {\_a} +c_{1}\right )}{x^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 116.627 (sec). Leaf size: 67473
DSolve[x^4 (y'[x])^3 -x^3 y[x] (y'[x])^2 - x^2 y[x]^2 y'[x]+x y[x]^3==1,y[x],x,IncludeSingularSolutions -> True]
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