1.86 problem 89

Internal problem ID [2722]

Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number: 89.
ODE order: 1.
ODE degree: 3.

CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {y-y^{\prime } x +x^{2} \left (y^{\prime }\right )^{3}=0} \end {gather*}

✓ Solution by Maple

Time used: 0.297 (sec). Leaf size: 5479

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

\begin{align*} y \relax (x ) = -\frac {1}{4 c_{1}} \\ y \relax (x ) = -\RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} x c_{1}+11 c_{1} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-39 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1} x +45 c_{1} x -\frac {\sqrt {16 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{6}-352 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{5}+3184 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{4}-15168 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3}+40176 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-56160 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )+32400 c_{1}^{4} x^{2}+26 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} c_{1}^{2} x -286 c_{1}^{2} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+982 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1}^{2} x -1010 c_{1}^{2} x +9}}{4 c_{1}}-\frac {3}{4 c_{1}} \\ y \relax (x ) = -\RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} x c_{1}+11 c_{1} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-39 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1} x +45 c_{1} x +\frac {\sqrt {16 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{6}-352 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{5}+3184 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{4}-15168 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3}+40176 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-56160 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )+32400 c_{1}^{4} x^{2}+26 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} c_{1}^{2} x -286 c_{1}^{2} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+982 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1}^{2} x -1010 c_{1}^{2} x +9}}{4 c_{1}}-\frac {3}{4 c_{1}} \\ y \relax (x ) = \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} x c_{1}+11 c_{1} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+39 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1} x +45 c_{1} x -\frac {\sqrt {16 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{6}+352 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{5}+3184 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{4}+15168 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3}+40176 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+56160 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )+32400 c_{1}^{4} x^{2}-26 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} c_{1}^{2} x -286 c_{1}^{2} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-982 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1}^{2} x -1010 c_{1}^{2} x +9}}{4 c_{1}}-\frac {3}{4 c_{1}} \\ y \relax (x ) = \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} x c_{1}+11 c_{1} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+39 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1} x +45 c_{1} x +\frac {\sqrt {16 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{6}+352 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{5}+3184 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{4}+15168 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3}+40176 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+56160 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )+32400 c_{1}^{4} x^{2}-26 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} c_{1}^{2} x -286 c_{1}^{2} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-982 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1}^{2} x -1010 c_{1}^{2} x +9}}{4 c_{1}}-\frac {3}{4 c_{1}} \\ y \relax (x ) = \frac {\RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} x c_{1}}{4}-\frac {13 c_{1} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}}{4}+\frac {51 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1} x}{4}-\frac {63 c_{1} x}{4}-\frac {\sqrt {4 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{6}-104 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{5}+1084 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{4}-5808 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3}+16956 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-25704 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )+15876 c_{1}^{4} x^{2}-20 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} c_{1}^{2} x +260 c_{1}^{2} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-988 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1}^{2} x +1036 c_{1}^{2} x +9}}{8 c_{1}}-\frac {3}{8 c_{1}} \\ y \relax (x ) = \frac {\RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} x c_{1}}{4}-\frac {13 c_{1} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}}{4}+\frac {51 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1} x}{4}-\frac {63 c_{1} x}{4}+\frac {\sqrt {4 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{6}-104 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{5}+1084 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{4}-5808 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3}+16956 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-25704 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )+15876 c_{1}^{4} x^{2}-20 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} c_{1}^{2} x +260 c_{1}^{2} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}-988 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}-24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (-216 c_{1}^{2} x -1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1}^{2} x +1036 c_{1}^{2} x +9}}{8 c_{1}}-\frac {3}{8 c_{1}} \\ y \relax (x ) = -\frac {\RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} x c_{1}}{4}-\frac {13 c_{1} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}}{4}-\frac {51 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1} x}{4}-\frac {63 c_{1} x}{4}-\frac {\sqrt {4 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{6}+104 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{5}+1084 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{4}+5808 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3}+16956 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+25704 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )+15876 c_{1}^{4} x^{2}+20 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} c_{1}^{2} x +260 c_{1}^{2} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+988 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1}^{2} x +1036 c_{1}^{2} x +9}}{8 c_{1}}-\frac {3}{8 c_{1}} \\ y \relax (x ) = -\frac {\RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} x c_{1}}{4}-\frac {13 c_{1} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}}{4}-\frac {51 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1} x}{4}-\frac {63 c_{1} x}{4}+\frac {\sqrt {4 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{6}+104 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{5}+1084 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{4}+5808 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3}+16956 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+25704 c_{1}^{4} x^{2} \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )+15876 c_{1}^{4} x^{2}+20 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{3} c_{1}^{2} x +260 c_{1}^{2} x \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right )^{2}+988 \RootOf \left (2 c_{1}^{2} x \,\textit {\_Z}^{4}+24 c_{1}^{2} x \,\textit {\_Z}^{3}+108 c_{1}^{2} x \,\textit {\_Z}^{2}+\left (216 c_{1}^{2} x +1\right ) \textit {\_Z} +162 c_{1}^{2} x -1\right ) c_{1}^{2} x +1036 c_{1}^{2} x +9}}{8 c_{1}}-\frac {3}{8 c_{1}} \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

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

Timed out