28.1.87 problem 90

Internal problem ID [4393]
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 : 90
Date solved : Monday, January 27, 2025 at 09:12:04 AM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.102 (sec). Leaf size: 571

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

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {3}}{9 x} \\ y \left (x \right ) &= \frac {\sqrt {3}}{9 x} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} +24 \left (\int _{}^{\textit {\_Z}}\frac {\left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}} \textit {\_a}}{36 \left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}} \textit {\_a}^{2}+\left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{2}/{3}}-24 \textit {\_a}^{2}-\left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}}+1}d \textit {\_a} \right )\right )}{x} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} -48 \left (\int _{}^{\textit {\_Z}}\frac {\left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}} \textit {\_a}}{i \left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{2}/{3}} \sqrt {3}+24 i \sqrt {3}\, \textit {\_a}^{2}-72 \left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}} \textit {\_a}^{2}-i \sqrt {3}+\left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{2}/{3}}-24 \textit {\_a}^{2}+2 \left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}}+1}d \textit {\_a} \right )\right )}{x} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} +48 \left (\int _{}^{\textit {\_Z}}\frac {\left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}} \textit {\_a}}{i \left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{2}/{3}} \sqrt {3}+24 i \sqrt {3}\, \textit {\_a}^{2}+72 \left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}} \textit {\_a}^{2}-i \sqrt {3}-\left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{2}/{3}}+24 \textit {\_a}^{2}-2 \left (24 \textit {\_a}^{3} \sqrt {81 \textit {\_a}^{2}-3}-216 \textit {\_a}^{4}+36 \textit {\_a}^{2}-1\right )^{{1}/{3}}-1}d \textit {\_a} \right )\right )}{x} \\ \end{align*}

✗ Solution by Mathematica

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

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

Timed out