Internal problem ID [1009]
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: 32.
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 {y}{y-2 x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 335
dsolve(diff(y(x),x)=y(x)/(y(x)-2*x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2}}{\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+x \\ y \relax (x ) = -\frac {\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+x -\frac {i \sqrt {3}\, \left (\frac {\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2}}{\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+x +\frac {i \sqrt {3}\, \left (\frac {\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 x^{2}}{\left (-12 c_{1}+8 x^{3}+4 \sqrt {-12 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.561 (sec). Leaf size: 441
DSolve[y'[x]==y[x]/(y[x]-2*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}+x \\ y(x)\to \frac {(-1)^{2/3} \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{-2} x^2}{\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}+x \\ y(x)\to \frac {x^2 \text {Root}\left [\text {$\#$1}^3-2\&,3\right ]}{\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}-\sqrt [3]{-\frac {1}{2}} \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}+x \\ y(x)\to 0 \\ y(x)\to \frac {i \left (\sqrt [3]{x^3}-x\right ) \left (\left (\sqrt {3}+i\right ) \sqrt [3]{x^3}+2 i x\right )}{2 x} \\ y(x)\to -\frac {i \left (\sqrt [3]{x^3}-x\right ) \left (\left (\sqrt {3}-i\right ) \sqrt [3]{x^3}-2 i x\right )}{2 x} \\ y(x)\to \sqrt [3]{x^3}+\frac {\left (x^3\right )^{2/3}}{x}+x \\ \end{align*}