Internal problem ID [10674]
Internal file name [OUTPUT/9622_Monday_June_06_2022_03_16_12_PM_56059788/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: 26.
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.
\[ \boxed {y y^{\prime }-y=-\frac {2 x}{9}+\frac {A}{\sqrt {x}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -9 y y^{\prime } \sqrt {x}+9 y \sqrt {x}-2 x^{\frac {3}{2}}+9 A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-9 y \sqrt {x}+2 x^{\frac {3}{2}}-9 A}{9 y \sqrt {x}} \end {array} \]
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: 134
dsolve(y(x)*diff(y(x),x)-y(x)=-2/9*x+A*x^(-1/2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \,3^{\frac {5}{6}} 2^{\frac {1}{3}} \left (-2 x^{\frac {3}{2}}+9 A \right )}{\sqrt {x}\, \left (\left (27 \tan \left (\operatorname {RootOf}\left (18 \,3^{\frac {5}{6}} 2^{\frac {1}{3}} \left (\int \frac {\left (\frac {A}{x^{\frac {3}{2}}}\right )^{\frac {2}{3}} \sqrt {x}}{-2 x^{\frac {3}{2}}+9 A}d x \right )+\ln \left (-8 \sqrt {3}\, \sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right )^{3}-8 \cos \left (\textit {\_Z} \right )^{4}-4 \sqrt {3}\, \sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right )+16 \cos \left (\textit {\_Z} \right )^{2}+1\right ) \sqrt {3}-12 \sqrt {3}\, c_{1} -12 \textit {\_Z} \right )\right )-9 \sqrt {3}\right ) \left (\frac {A}{x^{\frac {3}{2}}}\right )^{\frac {1}{3}}-6 \,2^{\frac {1}{3}} 3^{\frac {5}{6}}\right )} \]
✓ Solution by Mathematica
Time used: 1.355 (sec). Leaf size: 282
DSolve[y[x]*y'[x]-y[x]==-2/9*x+A*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\log \left (9 A^{2/3}+3 \sqrt [3]{6} \sqrt [3]{A} \sqrt {x}+6^{2/3} x\right )+2 \sqrt {3} \arctan \left (\frac {-\frac {6 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}-27}{27 \sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{6} \sqrt {x}}{\sqrt [3]{A}}+3}{3 \sqrt {3}}\right )+2 \log \left (\frac {1}{27} \left (27-\frac {3 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}\right )\right )=\log \left (\frac {1}{81} \left (\frac {6^{2/3} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )^2}{A^{2/3} y(x)^2}+\frac {9 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}+81\right )\right )+2 \log \left (3 \sqrt [3]{A}-\sqrt [3]{6} \sqrt {x}\right )+6 c_1,y(x)\right ] \]