Internal problem ID [10657]
Internal file name [OUTPUT/9605_Monday_June_06_2022_03_13_23_PM_32679015/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: 9.
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.
\[ \boxed {y y^{\prime }-y=A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-y=A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y+A \left ({\mathrm e}^{\frac {2 x}{A}}-1\right )}{y} \end {array} \]
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: 82
dsolve(y(x)*diff(y(x),x)-y(x)=A*(exp(2*x/A)-1),y(x), singsol=all)
\[ c_{1} +2 A \arctan \left (\frac {A -y \left (x \right )}{y \left (x \right ) \sqrt {\frac {{\mathrm e}^{\frac {2 x}{A}} A^{2}-\left (A -y \left (x \right )\right )^{2}}{y \left (x \right )^{2}}}}\right )+2 y \left (x \right ) \sqrt {\frac {{\mathrm e}^{\frac {2 x}{A}} A^{2}-\left (A -y \left (x \right )\right )^{2}}{y \left (x \right )^{2}}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==A*(Exp[2*x/A]-1),y[x],x,IncludeSingularSolutions -> True]
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