83.46.9 problem Ex 9 page 73

Internal problem ID [19574]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter V. Singular solutions
Problem number : Ex 9 page 73
Date solved : Tuesday, January 28, 2025 at 01:57:32 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} x^{2} {y^{\prime }}^{3}+y y^{\prime } \left (y+2 x \right )+y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.129 (sec). Leaf size: 1214

dsolve(x^2*diff(y(x),x)^3+y(x)*diff(y(x),x)*(2*x+y(x))+y(x)^2=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-6 \ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {-{\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{2}/{3}}+8 {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}} \textit {\_a} +8 \textit {\_a}^{2}+4 {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}}+44 \textit {\_a} -16}{\textit {\_a} \left (2 \textit {\_a} +3\right ) {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}}}d \textit {\_a} +6 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-12 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {i \sqrt {3}\, {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{2}/{3}}+8 i \sqrt {3}\, \textit {\_a}^{2}+44 i \sqrt {3}\, \textit {\_a} -16 i \sqrt {3}-{\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{2}/{3}}-16 {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}} \textit {\_a} +8 \textit {\_a}^{2}-8 {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}}+44 \textit {\_a} -16}{\textit {\_a} \left (2 \textit {\_a} +3\right ) {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}}}d \textit {\_a} +12 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-12 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {i \sqrt {3}\, {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{2}/{3}}+8 i \sqrt {3}\, \textit {\_a}^{2}+44 i \sqrt {3}\, \textit {\_a} -16 i \sqrt {3}+{\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{2}/{3}}+16 {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}} \textit {\_a} -8 \textit {\_a}^{2}+8 {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}}-44 \textit {\_a} +16}{\textit {\_a} \left (2 \textit {\_a} +3\right ) {\left (24 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a}^{2}+36 \sqrt {3}\, \sqrt {\frac {4 \textit {\_a}^{3}+24 \textit {\_a}^{2}+75 \textit {\_a} +32}{\textit {\_a}}}\, \textit {\_a} -80 \textit {\_a}^{3}-336 \textit {\_a}^{2}-708 \textit {\_a} -64\right )}^{{1}/{3}}}d \textit {\_a} +12 c_{1} \right ) x \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[x^2*D[y[x],x]^3+y[x]*D[y[x],x]*(2*x+y[x])+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

Timed out