Internal problem ID [1008]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable
Equations. Section 2.4 Page 68
Problem number: 31.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x +2 y}{y+2 x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.173 (sec). Leaf size: 385
dsolve(diff(y(x),x)=(x+2*y(x))/(2*x+y(x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x \left (c_{1}^{2} \left (\frac {\left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}{3 x c_{1}}+\frac {1}{x c_{1} \left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}\right )+c_{1}^{2}\right )}{c_{1}^{2}} \\ y \relax (x ) = \frac {x \left (c_{1}^{2} \left (-\frac {\left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}{6 x c_{1}}-\frac {1}{2 x c_{1} \left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}{3 x c_{1}}-\frac {1}{x c_{1} \left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}\right )}{2}\right )+c_{1}^{2}\right )}{c_{1}^{2}} \\ y \relax (x ) = \frac {x \left (c_{1}^{2} \left (-\frac {\left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}{6 x c_{1}}-\frac {1}{2 x c_{1} \left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}{3 x c_{1}}-\frac {1}{x c_{1} \left (27 x c_{1}+3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}\right )}{2}\right )+c_{1}^{2}\right )}{c_{1}^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.104 (sec). Leaf size: 338
DSolve[y'[x]==(x+2*y[x])/(2*x+y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}{3^{2/3}}-\frac {e^{2 c_1}}{\sqrt [3]{3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}+x \\ y(x)\to \left (-\frac {1}{3}\right )^{2/3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}+\frac {\sqrt [3]{-\frac {1}{3}} e^{2 c_1}}{\sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}+x \\ y(x)\to \frac {1}{3} \left (-\sqrt [3]{-3} \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}-\frac {(-3)^{2/3} e^{2 c_1}}{\sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}+3 x\right ) \\ \end{align*}