Internal problem ID [10693]
Internal file name [OUTPUT/9641_Monday_June_06_2022_03_17_26_PM_14824341/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: 45.
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.
\[ \boxed {y y^{\prime }-y=-\frac {6}{25} x -A \,x^{2}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-y=-\frac {6}{25} x -A \,x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y-\frac {6 x}{25}-A \,x^{2}}{y} \end {array} \]
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: 132
dsolve(y(x)*diff(y(x),x)-y(x)=-6/25*x-A*x^2,y(x), singsol=all)
\[ c_{1} +\frac {\left (2 x -5 y \left (x \right )\right ) \left (\int _{}^{-\frac {10 \sqrt {-x A}\, x}{-2 x +5 y \left (x \right )}}\frac {\left (\textit {\_a}^{2}-6\right )^{\frac {1}{6}}}{\textit {\_a}^{\frac {1}{3}}}d \textit {\_a} \right )-\frac {5 \,2^{\frac {5}{6}} \left (\frac {-50 A \,x^{3}-12 x^{2}+60 y \left (x \right ) x -75 y \left (x \right )^{2}}{\left (-2 x +5 y \left (x \right )\right )^{2}}\right )^{\frac {1}{6}} 5^{\frac {2}{3}} \sqrt {-x A}\, y \left (x \right )}{2 \left (-\frac {\sqrt {-x A}\, x}{-2 x +5 y \left (x \right )}\right )^{\frac {1}{3}}}}{2 x -5 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 2.139 (sec). Leaf size: 162
DSolve[y[x]*y'[x]-y[x]==-6/25*x-A*x^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {i \sqrt [6]{\frac {-2 x^2 (25 A x+6)+60 x y(x)-75 y(x)^2}{A x^3}} \left (25 A x^2-\frac {\sqrt [6]{2} \sqrt [3]{5} (2 x-5 y(x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{6},\frac {3}{2},-\frac {3 (2 x-5 y(x))^2}{50 A x^3}\right )}{\sqrt [6]{\frac {2 x^2 (25 A x+6)-60 x y(x)+75 y(x)^2}{A x^3}}}\right )}{5\ 2^{2/3} \sqrt {3} \sqrt [3]{5} \sqrt {A} x^{3/2}},y(x)\right ] \]