Internal problem ID [10792]
Internal file name [OUTPUT/9740_Monday_June_06_2022_04_51_00_PM_3336046/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. subsection 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 56.
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.
\[ \boxed {y y^{\prime }-\frac {a \left (\left (k +1\right ) x -1\right ) y}{x^{2}}=\frac {a^{2} \left (k +1\right ) \left (x -1\right )}{x^{2}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } x^{2}-y a k x -a^{2} k x -a x y+a^{2} k -a^{2} x +a y+a^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y a k x +a^{2} k x +a x y-a^{2} k +a^{2} x -a y-a^{2}}{x^{2} y} \end {array} \]
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 Looking for potential symmetries Looking for potential symmetries found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 143
dsolve(y(x)*diff(y(x),x)-a*((k+1)*x-1)*x^(-2)*y(x)=a^2*(k+1)*(x-1)*x^(-2),y(x), singsol=all)
\[ \frac {\left (\frac {a x}{-y \left (x \right ) x +a}\right )^{-\frac {1}{1+k}} x^{2} \left (\frac {\left (-1+x \right ) a +y \left (x \right ) x}{-y \left (x \right ) x +a}\right )^{\frac {1}{1+k}} {\mathrm e}^{\frac {-y \left (x \right ) x +a}{a \left (1+k \right ) x}} y \left (x \right )-\left (\int _{}^{\frac {a x}{-y \left (x \right ) x +a}}\left (\textit {\_a} -1\right )^{\frac {1}{1+k}} {\mathrm e}^{\frac {1}{\left (1+k \right ) \textit {\_a}}} \textit {\_a}^{-\frac {1}{1+k}}d \textit {\_a} -c_{1} \right ) \left (-y \left (x \right ) x +a \right )}{-y \left (x \right ) x +a} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*((k+1)*x-1)*x^(-2)*y[x]==a^2*(k+1)*(x-1)*x^(-2),y[x],x,IncludeSingularSolutions -> True]
Not solved