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`]]
\[ \boxed {y^{\prime }-\frac {x +2 y}{2 x +y}=0} \]
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 271
dsolve(diff(y(x),x)=(x+2*y(x))/(2*x+y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {x \left (\frac {c_{1} \left (\left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {2}{3}}+3\right )}{3 x \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}+c_{1}^{2}\right )}{c_{1}^{2}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {2}{3}}-6 x c_{1} \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}-3 i \sqrt {3}+3}{6 \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {2}{3}}+6 x c_{1} \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}-3 i \sqrt {3}-3}{6 \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 30.082 (sec). Leaf size: 382
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 \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}{2\ 3^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) e^{2 c_1}}{2 \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 -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {27 e^{4 c_1} x^2+e^{6 c_1}}-9 e^{2 c_1} x}}{2\ 3^{2/3}}+\frac {\left (1-i \sqrt {3}\right ) e^{2 c_1}}{2 \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 \\ \end{align*}