Internal problem ID [10651]
Internal file name [OUTPUT/9599_Monday_June_06_2022_03_13_14_PM_3277166/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: 3.
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.
\[ \boxed {y y^{\prime }-y=-\frac {2 x}{9}+A +\frac {B}{\sqrt {x}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -9 y y^{\prime } \sqrt {x}+9 A \sqrt {x}+9 y \sqrt {x}-2 x^{\frac {3}{2}}+9 B =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-9 A \sqrt {x}-9 y \sqrt {x}+2 x^{\frac {3}{2}}-9 B}{9 y \sqrt {x}} \end {array} \]
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: 90
dsolve(y(x)*diff(y(x),x)-y(x)=-2/9*x+A+B*x^(-1/2),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {9 \left (A \sqrt {x}+B -\frac {2 x^{\frac {3}{2}}}{9}\right ) A}{3 A \sqrt {x}+3 \operatorname {RootOf}\left (18 A^{3} \left (\int _{}^{\textit {\_Z}}\frac {1}{-2 \textit {\_a}^{3} B^{2}+9 \textit {\_a} \,A^{3}-9 A^{3}}d \textit {\_a} \right )-9 A \left (\int \frac {1}{9 x A -2 x^{2}+9 B \sqrt {x}}d x \right )+2 c_{1} \right ) B} \]
✓ Solution by Mathematica
Time used: 8.154 (sec). Leaf size: 415
DSolve[y[x]*y'[x]-y[x]==-2/9*x+A+B*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [6 \text {RootSum}\left [8 \text {$\#$1}^6-72 \text {$\#$1}^4 A-36 \text {$\#$1}^4 y(x)-72 \text {$\#$1}^3 B+162 \text {$\#$1}^2 A^2+162 \text {$\#$1}^2 A y(x)+54 \text {$\#$1}^2 y(x)^2+324 \text {$\#$1} A B+162 \text {$\#$1} B y(x)-81 A y(x)^2+162 B^2-27 y(x)^3\&,\frac {-2 \text {$\#$1}^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+9 \text {$\#$1} A \log \left (\sqrt {x}-\text {$\#$1}\right )+9 B \log \left (\sqrt {x}-\text {$\#$1}\right )+9 \text {$\#$1} y(x) \log \left (\sqrt {x}-\text {$\#$1}\right )}{8 \text {$\#$1}^5-48 \text {$\#$1}^3 A-24 \text {$\#$1}^3 y(x)-36 \text {$\#$1}^2 B+54 \text {$\#$1} A^2+54 \text {$\#$1} A y(x)+18 \text {$\#$1} y(x)^2+54 A B+27 B y(x)}\&\right ]+\int _1^{y(x)}\left (\frac {162 K[1]}{8 x^3-72 A x^2-36 K[1] x^2-72 B x^{3/2}+162 A^2 x+54 K[1]^2 x+162 A K[1] x+324 A B \sqrt {x}+162 B K[1] \sqrt {x}-27 K[1]^3+162 B^2-81 A K[1]^2}+\frac {162 K[1]}{-8 x^3+72 A x^2+36 K[1] x^2+72 B x^{3/2}-162 A^2 x-54 K[1]^2 x-162 A K[1] x-324 A B \sqrt {x}-162 B K[1] \sqrt {x}+27 K[1]^3-162 B^2+81 A K[1]^2}\right )dK[1]=c_1,y(x)\right ] \]