60.2.168 problem 744

Internal problem ID [10755]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 744
Date solved : Tuesday, January 28, 2025 at 05:11:38 PM
CAS classification : [_rational]

\begin{align*} y^{\prime }&=\frac {x}{-y+x^{4}+2 x^{2} y^{2}+y^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 507

dsolve(diff(y(x),x) = x/(-y(x)+x^4+2*x^2*y(x)^2+y(x)^4),y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {3 c_{1}^{4} x^{2}+24 c_{1}^{2} x^{4}+48 x^{6}+3 c_{1}^{3}+108 c_{1} x^{2}+81}\right )^{{1}/{3}}}{6}+\frac {c_{1}^{2}-12 x^{2}}{6 \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {3 c_{1}^{4} x^{2}+24 c_{1}^{2} x^{4}+48 x^{6}+3 c_{1}^{3}+108 c_{1} x^{2}+81}\right )^{{1}/{3}}}-\frac {c_{1}}{6} \\ y &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{{2}/{3}}+\frac {c_{1} \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{{1}/{3}}}{6}+\left (x^{2}-\frac {c_{1}^{2}}{12}\right ) \left (i \sqrt {3}-1\right )}{\left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{{1}/{3}}} \\ y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{{2}/{3}}}{12}-\frac {c_{1} \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{{1}/{3}}}{6}+\left (1+i \sqrt {3}\right ) \left (x^{2}-\frac {c_{1}^{2}}{12}\right )}{\left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 13.804 (sec). Leaf size: 564

DSolve[D[y[x],x] == x/(x^4 - y[x] + 2*x^2*y[x]^2 + y[x]^4),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{6 \sqrt [3]{2}}+\frac {2^{2/3} \left (-3 x^2+c_1{}^2\right )}{3 \sqrt [3]{36 c_1 x^2+3 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-27+4 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{12 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (3 x^2-c_1{}^2\right )}{3 \sqrt [3]{72 c_1 x^2+6 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-54+8 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (3 x^2-c_1{}^2\right )}{3 \sqrt [3]{72 c_1 x^2+6 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-54+8 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to -i \sqrt {x^2} \\ y(x)\to i \sqrt {x^2} \\ \end{align*}