15.8.6 problem 6

Internal problem ID [3009]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 6
Date solved : Monday, January 27, 2025 at 07:07:41 AM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 536

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

\begin{align*} y &= \frac {12 x^{4}-4 x^{2}+{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{2}/{3}}}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{1}/{3}} \left (6 x^{2}-2\right )} \\ y &= \frac {\left (-1-i \sqrt {3}\right ) {\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{1}/{3}}+\frac {12 x^{2} \left (x^{2}-\frac {1}{3}\right ) \left (i \sqrt {3}-1\right )}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{1}/{3}}}}{12 x^{2}-4} \\ y &= \frac {\frac {\left (i \sqrt {3}-1\right ) {\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{2}/{3}}}{4}+3 \left (-1-i \sqrt {3}\right ) x^{2} \left (x^{2}-\frac {1}{3}\right )}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{{1}/{3}} \left (3 x^{2}-1\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.190 (sec). Leaf size: 723

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

\begin{align*} y(x)\to \frac {12 x^4-4 x^2+\left (27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1\right ){}^{2/3}}{2 \left (3 x^2-1\right ) \sqrt [3]{27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1}} \\ y(x)\to \frac {-12 i \left (\sqrt {3}-i\right ) x^4+\left (4+4 i \sqrt {3}\right ) x^2+i \left (\sqrt {3}+i\right ) \left (27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1\right ){}^{2/3}}{4 \left (3 x^2-1\right ) \sqrt [3]{27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1}} \\ y(x)\to \frac {12 i \left (\sqrt {3}+i\right ) x^4+\left (4-4 i \sqrt {3}\right ) x^2-i \left (\sqrt {3}-i\right ) \left (27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1\right ){}^{2/3}}{4 \left (3 x^2-1\right ) \sqrt [3]{27 x^8-18 x^6+3 (1+36 c_1) x^4-72 c_1 x^2+\sqrt {\left (3 x^2-1\right )^3 \left (27 x^{10}-9 x^8+8 (-8+27 c_1) x^6-72 c_1 x^4+432 c_1{}^2 x^2-144 c_1{}^2\right )}+12 c_1}} \\ \end{align*}