48.4.6 problem Problem 3.7

Internal problem ID [7548]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page 218
Problem number : Problem 3.7
Date solved : Monday, January 27, 2025 at 03:05:13 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 357

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

\begin{align*} y &= -\frac {12^{{1}/{3}} \left (-{\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+x^{2} 12^{{1}/{3}} \left (x^{2}-c_{1} \right )\right )}{6 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\ y &= -\frac {2^{{2}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x^{2} \left (x^{2}-c_{1} \right )\right ) 3^{{1}/{3}}}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\ y &= \frac {2^{{2}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x^{2} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) \left (x^{2}-c_{1} \right )\right ) 3^{{1}/{3}}}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{{1}/{3}} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 45.878 (sec). Leaf size: 358

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

\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (-x^3+c_1 x\right )}{\sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}+\frac {\sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{3 \sqrt [3]{2} x} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-c_1 x\right )}{2^{2/3} \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{6 \sqrt [3]{2} x} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-c_1 x\right )}{2^{2/3} \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{6 \sqrt [3]{2} x} \\ \end{align*}