Internal problem ID [8621]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 285.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {\left (4 y^{2}+2 y x +3 x^{2}\right ) y^{\prime }+y^{2}+6 y x=-2 x^{2}} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 432
dsolve((4*y(x)^2+2*x*y(x)+3*x^2)*diff(y(x),x)+y(x)^2+6*x*y(x)+2*x^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {1}{3}}-\frac {11 c_{1}^{2} x^{2}}{\left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {1}{3}}}-c_{1} x}{4 c_{1}} \\ y \left (x \right ) &= -\frac {11 i \sqrt {3}\, c_{1}^{2} x^{2}+i \sqrt {3}\, \left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {2}{3}}-11 c_{1}^{2} x^{2}+2 c_{1} x \left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {1}{3}}+\left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {2}{3}}}{8 \left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {11 i \sqrt {3}\, c_{1}^{2} x^{2}+i \sqrt {3}\, \left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {2}{3}}+11 c_{1}^{2} x^{2}-2 c_{1} x \left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {1}{3}}-\left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {2}{3}}}{8 \left (c_{1}^{3} x^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 c_{1}^{3} x^{3}+16}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 45.643 (sec). Leaf size: 612
DSolve[(4*y[x]^2+2*x*y[x]+3*x^2)*y'[x]+y[x]^2+6*x*y[x]+2*x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} \left (\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}-\frac {11 x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-x\right ) \\ y(x)\to \frac {1}{16} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}+\frac {22 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-4 x\right ) \\ y(x)\to \frac {1}{16} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}+\frac {22 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-4 x\right ) \\ y(x)\to \frac {1}{4} \left (\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}-\frac {11 x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-x\right ) \\ y(x)\to \frac {1}{8} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}+\frac {11 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-2 x\right ) \\ y(x)\to \frac {1}{8} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}+\frac {11 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-2 x\right ) \\ \end{align*}