29.36.6 problem 1070

Internal problem ID [5633]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 36
Problem number : 1070
Date solved : Monday, January 27, 2025 at 12:41:10 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} x^{4} {y^{\prime }}^{3}-x^{3} y {y^{\prime }}^{2}-x^{2} y^{2} y^{\prime }+x y^{3}&=1 \end{align*}

✓ Solution by Maple

Time used: 0.510 (sec). Leaf size: 686

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^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}}}{4 x} \\ y \left (x \right ) &= -\frac {3 \,2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{8 x} \\ y \left (x \right ) &= \frac {3 \,2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\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 )^{{1}/{3}}}{8 \,2^{{2}/{3}} \textit {\_a}^{2}+2^{{1}/{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{2}/{3}}+4 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{1}/{3}}}d \textit {\_a} \right )+c_{1} \right )}{x^{{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 )^{{1}/{3}}}{4 i \sqrt {3}\, 2^{{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 )^{{1}/{3}}+4 \,2^{{2}/{3}} \textit {\_a}^{2}-2^{{1}/{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{2}/{3}}+2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{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 )^{{1}/{3}}}{4 i \sqrt {3}\, 2^{{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 )^{{1}/{3}}+4 \,2^{{2}/{3}} \textit {\_a}^{2}-2^{{1}/{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{2}/{3}}+2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{1}/{3}}}d \textit {\_a} \right )-c_{1} \right )}{x^{{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 )^{{1}/{3}}}{4 i \sqrt {3}\, 2^{{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 )^{{1}/{3}}-4 \,2^{{2}/{3}} \textit {\_a}^{2}+2^{{1}/{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{2}/{3}}-2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{1}/{3}}}d \textit {\_a} \right )+3 \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{1}/{3}}}{4 i \sqrt {3}\, 2^{{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 )^{{1}/{3}}-4 \,2^{{2}/{3}} \textit {\_a}^{2}+2^{{1}/{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{2}/{3}}-2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{{1}/{3}}}d \textit {\_a} \right )+\ln \left (x \right )-c_{1} \right )}{x^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 89.135 (sec). Leaf size: 67473

DSolve[x^4 (D[y[x],x])^3 -x^3 y[x] (D[y[x],x])^2 - x^2 y[x]^2 D[y[x],x]+x y[x]^3==1,y[x],x,IncludeSingularSolutions -> True]
 

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