Internal problem ID [10656]
Internal file name [OUTPUT/9604_Monday_June_06_2022_03_13_22_PM_9151881/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: 8.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-y=A +B \,{\mathrm e}^{-\frac {2 x}{A}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-y=A +B \,{\mathrm e}^{-\frac {2 x}{A}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y+A +B \,{\mathrm e}^{-\frac {2 x}{A}}}{y} \end {array} \]
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: 76
dsolve(y(x)*diff(y(x),x)-y(x)=A+B*exp(-2*x/A),y(x), singsol=all)
\[ c_{1} -2 \arctan \left (\frac {y \left (x \right )+A}{y \left (x \right ) \sqrt {\frac {-A B \,{\mathrm e}^{-\frac {2 x}{A}}-\left (y \left (x \right )+A \right )^{2}}{y \left (x \right )^{2}}}}\right ) A -2 \sqrt {\frac {-A B \,{\mathrm e}^{-\frac {2 x}{A}}-\left (y \left (x \right )+A \right )^{2}}{y \left (x \right )^{2}}}\, y \left (x \right ) = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==A+B*Exp[-2*x/A],y[x],x,IncludeSingularSolutions -> True]
Not solved