Internal problem ID [5242]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 4. Equations of first order and first degree (Variable separable). Supplemetary
problems. Page 22
Problem number: 30.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class D`], _rational]
\[ \boxed {y^{2} \left (x^{2}+2\right )+\left (x^{3}+y^{3}\right ) \left (-x y^{\prime }+y\right )=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 659
dsolve(y(x)^2*(x^2+2)+(x^3+y(x)^3)*(y(x)-x*diff(y(x),x))=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {6 \ln \left (x \right ) x^{2}+6 c_{1} x^{2}+\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}-6}{3 \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (-1-i \sqrt {3}\right ) \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}}{6}+\left (i \sqrt {3}-1\right ) \left (c_{1} x^{2}+\ln \left (x \right ) x^{2}-1\right )}{\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {2}{3}}}{6}+\left (-1-i \sqrt {3}\right ) \left (c_{1} x^{2}+\ln \left (x \right ) x^{2}-1\right )}{\left (27 x^{3}+3 \sqrt {-24 c_{1}^{3} x^{6}-72 c_{1}^{2} x^{6} \ln \left (x \right )+72 c_{1}^{2} x^{4}-72 c_{1} x^{6} \ln \left (x \right )^{2}+144 c_{1} x^{4} \ln \left (x \right )-72 c_{1} x^{2}-24 \ln \left (x \right )^{3} x^{6}+72 \ln \left (x \right )^{2} x^{4}-72 \ln \left (x \right ) x^{2}+24+81 x^{6}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 54.35 (sec). Leaf size: 396
DSolve[y[x]^2*(x^2+2)+(x^3+y[x]^3)*(y[x]-x*y'[x])==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {6 x^2 \log (x)+6 c_1 x^2+3^{2/3} \left (9 x^3+\frac {1}{3} \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}\right ){}^{2/3}-6}{3 \sqrt [3]{3} \sqrt [3]{9 x^3+\frac {1}{3} \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^3+\frac {1}{3} \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}}{2\ 3^{2/3}}-\frac {i \sqrt [3]{2} \left (\sqrt {3}-i\right ) \left (x^2 \log (x)+c_1 x^2-1\right )}{\sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}} \\ y(x)\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (x^2 \log (x)+c_1 x^2-1\right )}{\sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x^3+2 \sqrt {729 x^6+\left (-6 x^2 \log (x)-6 c_1 x^2+6\right ){}^3}}}{6 \sqrt [3]{2}} \\ \end{align*}