Internal problem ID [9950]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. Form \(y y'-y=f(x)\). subsection 1.3.1-2. Solvable
equations and their solutions
Problem number: 44.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y y^{\prime }-y-A \,x^{2}+\frac {9}{625 A}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 195
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^2-9/625*A^(-1),y(x), singsol=all)
\[ c_{1}-\frac {125 \left (-\frac {2 \left (\frac {\left (25 x A +3\right )^{\frac {3}{2}}}{50 x A -125 A y \relax (x )+6}\right )^{\frac {1}{3}} \left (6+\left (50 x -125 y \relax (x )\right ) A \right ) \left (\int _{}^{\frac {2 \left (25 x A +3\right )^{\frac {3}{2}}}{6+\left (50 x -125 y \relax (x )\right ) A}}\frac {\left (\textit {\_a}^{2}-6\right )^{\frac {1}{6}}}{\textit {\_a}^{\frac {1}{3}}}d \textit {\_a} \right )}{125}+2^{\frac {5}{6}} \left (\frac {-54+31250 A^{3} x^{3}+\left (3750 x^{2}+37500 x y \relax (x )-46875 y \relax (x )^{2}\right ) A^{2}+\left (-450 x +4500 y \relax (x )\right ) A}{\left (50 x A -125 A y \relax (x )+6\right )^{2}}\right )^{\frac {1}{6}} A y \relax (x ) \sqrt {25 x A +3}\right )}{\left (\frac {\left (25 x A +3\right )^{\frac {3}{2}}}{6+\left (50 x -125 y \relax (x )\right ) A}\right )^{\frac {1}{3}} \left (12+\left (100 x -250 y \relax (x )\right ) A \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.458 (sec). Leaf size: 198
DSolve[y[x]*y'[x]-y[x]==A*x^2-9/625*A^(-1),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [6]{\frac {46875 A^2 y(x)^2-1500 A (25 A x+3) y(x)-2 (25 A x-3) (25 A x+3)^2}{(25 A x+3)^3}} \left (\frac {(-125 A y(x)+50 A x+6) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {3 (50 A x-125 A y(x)+6)^2}{2 (25 A x+3)^3}\right )}{\sqrt [3]{2} \sqrt {3} (25 A x+3)^{3/2} \sqrt [6]{\frac {-46875 A^2 y(x)^2+1500 A (25 A x+3) y(x)+2 (25 A x-3) (25 A x+3)^2}{(25 A x+3)^3}}}+\frac {\sqrt {25 A x+3}}{\sqrt {6}}\right )}{\sqrt [6]{2}}+c_1=0,y(x)\right ] \]