Internal problem ID [10692]
Internal file name [OUTPUT/9640_Monday_June_06_2022_03_17_25_PM_32058918/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-y=A \,x^{2}-\frac {9}{625 A}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 625 A^{2} x^{2}-625 y y^{\prime } A +625 A y-9=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-625 A^{2} x^{2}-625 A y+9}{625 A y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 186
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^2-9/625*A^(-1),y(x), singsol=all)
\[ \frac {-\frac {125 \,2^{\frac {5}{6}} \left (\frac {-46875 A^{2} y \left (x \right )^{2}+\left (37500 A^{2} x +4500 A \right ) y \left (x \right )+31250 \left (x A -\frac {3}{25}\right ) \left (x A +\frac {3}{25}\right )^{2}}{\left (50 x A -125 y \left (x \right ) A +6\right )^{2}}\right )^{\frac {1}{6}} A y \left (x \right ) \sqrt {25 x A +3}}{2}+50 \left (x A -\frac {5 y \left (x \right ) A}{2}+\frac {3}{25}\right ) \left (\frac {\left (25 x A +3\right )^{\frac {3}{2}}}{50 x A -125 y \left (x \right ) A +6}\right )^{\frac {1}{3}} \left (\int _{}^{-\frac {2 \left (25 x A +3\right )^{\frac {3}{2}}}{125 y \left (x \right ) A -50 x A -6}}\frac {\left (\textit {\_a}^{2}-6\right )^{\frac {1}{6}}}{\textit {\_a}^{\frac {1}{3}}}d \textit {\_a} +c_{1} \right )}{\left (\frac {\left (25 x A +3\right )^{\frac {3}{2}}}{50 x A -125 y \left (x \right ) A +6}\right )^{\frac {1}{3}} \left (50 x A -125 y \left (x \right ) A +6\right )} = 0 \]
✓ Solution by Mathematica
Time used: 2.438 (sec). Leaf size: 198
DSolve[y[x]*y'[x]-y[x]==A*x^2-9/625*A^(-1),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [6]{\frac {46875 A^2 y(x)^2-1500 A (25 A x+3) y(x)-2 (25 A x-3) (25 A x+3)^2}{(25 A x+3)^3}} \left (\frac {(-125 A y(x)+50 A x+6) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},\frac {3 (50 A x-125 A y(x)+6)^2}{2 (25 A x+3)^3}\right )}{\sqrt [3]{2} \sqrt {3} (25 A x+3)^{3/2} \sqrt [6]{\frac {-46875 A^2 y(x)^2+1500 A (25 A x+3) y(x)+2 (25 A x-3) (25 A x+3)^2}{(25 A x+3)^3}}}+\frac {\sqrt {25 A x+3}}{\sqrt {6}}\right )}{\sqrt [6]{2}}+c_1=0,y(x)\right ] \]