Internal problem ID [4476]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson
10
Problem number: Recognizable Exact Differential equations. Integrating factors. Exercise
10.1, page 90.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {2 x y+\left (x^{2}+y^{2}\right ) y^{\prime }=-x^{2}} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 321
dsolve((2*x*y(x)+x^2)+(x^2+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\left (i \sqrt {3}-1\right ) \sqrt {c_{1}}\, x^{2}}{\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}}{4 \left (4-4 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 23.867 (sec). Leaf size: 597
DSolve[(2*x*y[x]+x^2)+(x^2+y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\ y(x)\to \frac {2 \sqrt [3]{-2} x^2+(-2)^{2/3} \left (\sqrt {5} \sqrt {x^6}-x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\ y(x)\to \frac {\left (2 \sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}-2 \sqrt [3]{2} x^2}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\ y(x)\to \frac {\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x^2+\left (-1-i \sqrt {3}\right ) \left (2 \sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}}{4 \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\ \end{align*}