Internal problem ID [10659]
Internal file name [OUTPUT/9607_Monday_June_06_2022_03_13_26_PM_78895175/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: 11.
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 B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-y=-\frac {2 x}{9}+6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 108 A^{3}+54 A^{2} \sqrt {x}-9 y y^{\prime } \sqrt {x}+9 y \sqrt {x}-2 x^{\frac {3}{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-108 A^{3}-54 A^{2} \sqrt {x}-9 y \sqrt {x}+2 x^{\frac {3}{2}}}{9 y \sqrt {x}} \end {array} \]
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: 354
dsolve(y(x)*diff(y(x),x)-y(x)=-2/9*x+6*A^2*(1+2*A*x^(-1/2)),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-108 A^{3}-54 A^{2} \sqrt {x}+2 x^{\frac {3}{2}}}{3 \,{\mathrm e}^{\operatorname {RootOf}\left (36 A^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+18 A^{2} {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )+108 A^{2} c_{1} {\mathrm e}^{\textit {\_Z}}+36 A^{2} {\mathrm e}^{\textit {\_Z}} \textit {\_Z} +6 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )+18 A \sqrt {x}\, c_{1} {\mathrm e}^{\textit {\_Z}}+6 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}} \textit {\_Z} +108 A^{2} {\mathrm e}^{\textit {\_Z}}-18 A \sqrt {x}\, {\mathrm e}^{\textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} x \ln \left (2\right )-{\mathrm e}^{\textit {\_Z}} x \ln \left (\frac {\left (3 A -\sqrt {x}\right ) \left (6 A -\sqrt {x}\right ) \left (36 A^{2}-x \right )}{\left (9 A^{2}-x \right ) \left (6 A +\sqrt {x}\right ) \left (3 A +\sqrt {x}\right ) \left ({\mathrm e}^{\textit {\_Z}}+9\right )^{2}}\right )-6 c_{1} x \,{\mathrm e}^{\textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} x +324 A^{2}+54 A \sqrt {x}-18 x \right )} A +9 A +3 \sqrt {x}} \]
✓ Solution by Mathematica
Time used: 12.331 (sec). Leaf size: 488
DSolve[y[x]*y'[x]-y[x]==-2/9*x+6*A^2*(1+2*A*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2^{2/3} \left (\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{\sqrt [3]{A^3}}+54\right ) \left (\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+9 \sqrt {x} y(x)}{\sqrt [3]{A^3} y(x)}+27\right ) \left (-\frac {\left (3 \left (3 \sqrt [3]{A^3}+\sqrt {x}\right ) y(x)+2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2\right ) \log \left (\frac {1}{27} 2^{2/3} \left (\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{\sqrt [3]{A^3}}+54\right )\right )}{9 \sqrt [3]{A^3} y(x)}+\left (\frac {2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)}{9 \sqrt [3]{A^3} y(x)}+1\right ) \log \left (\frac {1}{27} 2^{2/3} \left (\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+9 \sqrt {x} y(x)}{\sqrt [3]{A^3} y(x)}+27\right )\right )-3\right )}{6561 \left (\frac {\left (2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)\right )^3}{729 A^3 y(x)^3}+\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{9 \sqrt [3]{A^3}}-2\right )}=\frac {2^{2/3} \left (A^3\right )^{2/3} \left (2 \text {arctanh}\left (\frac {1}{3}-\frac {2 \sqrt {x}}{9 A}\right )+\frac {9 A}{3 A+\sqrt {x}}\right )}{9 A^2}+c_1,y(x)\right ] \]