Internal problem ID [5027]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems.
page 12
Problem number: 31.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {y^{\prime }-\frac {2 y-x +5}{2 x -y-4}=0} \]
✓ Solution by Maple
Time used: 0.313 (sec). Leaf size: 182
dsolve(diff(y(x),x)=(2*y(x)-x+5)/(2*x-y(x)-4),y(x), singsol=all)
\[ y \left (x \right ) = -2-\frac {\left (x -1\right ) \left (c_{1}^{2} \left (-\frac {\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (x -1\right )^{2}-1}+27 c_{1} \left (x -1\right )\right )^{\frac {1}{3}}}{6 c_{1} \left (x -1\right )}-\frac {1}{2 c_{1} \left (x -1\right ) \left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (x -1\right )^{2}-1}+27 c_{1} \left (x -1\right )\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (x -1\right )^{2}-1}+27 c_{1} \left (x -1\right )\right )^{\frac {1}{3}}}{3 c_{1} \left (x -1\right )}-\frac {1}{c_{1} \left (x -1\right ) \left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (x -1\right )^{2}-1}+27 c_{1} \left (x -1\right )\right )^{\frac {1}{3}}}\right )}{2}\right )+c_{1}^{2}\right )}{c_{1}^{2}} \]
✓ Solution by Mathematica
Time used: 60.161 (sec). Leaf size: 629
DSolve[y'[x]==(2*y[x]-x+5)/(2*x-y[x]-4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2 (x-2)+\frac {3 (x-1)}{\frac {1}{\sqrt [3]{-e^{\frac {3 c_1}{4}} (x-1)^4+2 e^{\frac {3 c_1}{8}} (x-1)^2+\sqrt {e^{\frac {3 c_1}{8}} (x-1)^2 \left (-1+e^{\frac {3 c_1}{8}} (x-1)^2\right ){}^3}-1}}-\frac {\sqrt [3]{-e^{\frac {3 c_1}{4}} (x-1)^4+2 e^{\frac {3 c_1}{8}} (x-1)^2+\sqrt {e^{\frac {3 c_1}{8}} (x-1)^2 \left (-1+e^{\frac {3 c_1}{8}} (x-1)^2\right ){}^3}-1}}{(x-1)^2 \cosh \left (\frac {3 c_1}{8}\right )+(x-1)^2 \sinh \left (\frac {3 c_1}{8}\right )-1}-1} \\ y(x)\to 2 \left (x+\frac {3 (x-1)}{\frac {-1-i \sqrt {3}}{\sqrt [3]{-e^{\frac {3 c_1}{4}} (x-1)^4+2 e^{\frac {3 c_1}{8}} (x-1)^2+\sqrt {e^{\frac {3 c_1}{8}} (x-1)^2 \left (-1+e^{\frac {3 c_1}{8}} (x-1)^2\right ){}^3}-1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-e^{\frac {3 c_1}{4}} (x-1)^4+2 e^{\frac {3 c_1}{8}} (x-1)^2+\sqrt {e^{\frac {3 c_1}{8}} (x-1)^2 \left (-1+e^{\frac {3 c_1}{8}} (x-1)^2\right ){}^3}-1}}{(x-1)^2 \cosh \left (\frac {3 c_1}{8}\right )+(x-1)^2 \sinh \left (\frac {3 c_1}{8}\right )-1}-2}-2\right ) \\ y(x)\to 2 \left (x+\frac {3 (x-1)}{\frac {i \left (\sqrt {3}+i\right )}{\sqrt [3]{-e^{\frac {3 c_1}{4}} (x-1)^4+2 e^{\frac {3 c_1}{8}} (x-1)^2+\sqrt {e^{\frac {3 c_1}{8}} (x-1)^2 \left (-1+e^{\frac {3 c_1}{8}} (x-1)^2\right ){}^3}-1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-e^{\frac {3 c_1}{4}} (x-1)^4+2 e^{\frac {3 c_1}{8}} (x-1)^2+\sqrt {e^{\frac {3 c_1}{8}} (x-1)^2 \left (-1+e^{\frac {3 c_1}{8}} (x-1)^2\right ){}^3}-1}}{(x-1)^2 \cosh \left (\frac {3 c_1}{8}\right )+(x-1)^2 \sinh \left (\frac {3 c_1}{8}\right )-1}-2}-2\right ) \\ \end{align*}