Internal problem ID [10662]
Internal file name [OUTPUT/9610_Monday_June_06_2022_03_13_35_PM_29052279/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: 14.
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.
\[ \boxed {y y^{\prime }-y=\frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-y=\frac {4}{9} x +2 A \,x^{2}+2 A^{2} x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y+\frac {4 x}{9}+2 A \,x^{2}+2 A^{2} x^{3}}{y} \end {array} \]
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: 177
dsolve(y(x)*diff(y(x),x)-y(x)=4/9*x+2*A*x^2+2*A^2*x^3,y(x), singsol=all)
\[ -\frac {27 \left (A y \left (x \right ) 3^{\frac {1}{4}} \left (x A +\frac {1}{3}\right ) {\left (\frac {A \left (9 A^{2} x^{2}-9 y \left (x \right ) A +9 x A +2\right ) \left (3 A \,x^{2}+3 y \left (x \right )+x \right )}{\left (-9 y \left (x \right ) A +3 x A +1\right )^{2}}\right )}^{\frac {1}{4}}-\frac {\left (\int _{}^{\frac {\left (3 x A +1\right )^{2}}{-9 y \left (x \right ) A +3 x A +1}}\frac {\left (\textit {\_a}^{2}-1\right )^{\frac {1}{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +c_{1} \right ) \left (\frac {1}{3}-3 y \left (x \right ) A +x A \right ) \sqrt {\frac {\left (3 x A +1\right )^{2}}{-9 y \left (x \right ) A +3 x A +1}}}{9}\right )}{\sqrt {\frac {\left (3 x A +1\right )^{2}}{-9 y \left (x \right ) A +3 x A +1}}\, \left (-9 y \left (x \right ) A +3 x A +1\right )} = 0 \]
✓ Solution by Mathematica
Time used: 3.439 (sec). Leaf size: 170
DSolve[y[x]*y'[x]-y[x]==4/9*x+2*A*x^2+2*A^2*x^3,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\sqrt [4]{\frac {(-9 A y(x)+3 A x+1)^2}{(3 A x+1)^4}-1} \left (\frac {(-9 A y(x)+3 A x+1) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {(3 A x-9 A y(x)+1)^2}{(3 A x+1)^4}\right )}{2 \sqrt [4]{3} (3 A x+1) \sqrt {(3 A x+1)^2} \sqrt [4]{\frac {A \left (6 (3 A x+1) y(x)-27 A y(x)^2+x (3 A x+2) (3 A x+1)^2\right )}{(3 A x+1)^4}}}+\sqrt {(3 A x+1)^2}\right )+c_1=0,y(x)\right ] \]