Internal problem ID [4289]
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]]
\[ \boxed {x^{4} {y^{\prime }}^{3}-x^{3} y {y^{\prime }}^{2}-x^{2} y^{2} y^{\prime }+x y^{3}=1} \]
✓ Solution by Maple
Time used: 0.672 (sec). Leaf size: 698
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 \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4 x} \\ y \left (x \right ) &= -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{8 x} \\ y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{8 x} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+6 \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{8 \,2^{\frac {2}{3}} \textit {\_a}^{2}+2^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}+4 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )+c_{1} \right )}{x^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (3 i \sqrt {3}\, \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 i \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i \textit {\_a} \sqrt {3}\, \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^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )+\ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 i \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i \textit {\_a} \sqrt {3}\, \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^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} \right )}{x^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (3 i \sqrt {3}\, \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 i \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i \textit {\_a} \sqrt {3}\, \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^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}-2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )+\ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 i \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i \textit {\_a} \sqrt {3}\, \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^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}-2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} \right )}{x^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 84.141 (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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