Internal problem ID [10649]
Internal file name [OUTPUT/9597_Monday_June_06_2022_03_13_08_PM_16682338/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: 1.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y y^{\prime }-y=A} \]
Integrating both sides gives \begin {align*} \int \frac {y}{y +A}d y &= x +c_{1}\\ y -A \ln \left (y +A \right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-A \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {A +c_{1} +x}{A}}}{A}\right )+1\right )\\ &=-A \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1} {\mathrm e}^{-\frac {x}{A}}}{A c_{1}}\right )+1\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -A \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1} {\mathrm e}^{-\frac {x}{A}}}{A c_{1}}\right )+1\right ) \\ \end{align*}
Verification of solutions
\[ y = -A \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1} {\mathrm e}^{-\frac {x}{A}}}{A c_{1}}\right )+1\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-y=A \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y+A}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } y}{y+A}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } y}{y+A}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y-A \ln \left (y+A \right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-A \left (\mathit {LambertW}\left (-\frac {{\mathrm e}^{-\frac {A +c_{1} +x}{A}}}{A}\right )+1\right ) \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 30
dsolve(y(x)*diff(y(x),x)-y(x)=A,y(x), singsol=all)
\[ y \left (x \right ) = -A \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {-A -c_{1} -x}{A}}}{A}\right )+1\right ) \]
✓ Solution by Mathematica
Time used: 60.032 (sec). Leaf size: 28
DSolve[y[x]*y'[x]-y[x]==A,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -A \left (1+W\left (-\frac {e^{-\frac {A+x+c_1}{A}}}{A}\right )\right ) \]