Internal problem ID [6147]
Internal file name [OUTPUT/5395_Sunday_June_05_2022_03_36_10_PM_98519124/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page
12
Problem number: 1(i).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {x y y^{\prime }-y=-1} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y -1}{x y} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(y)=\frac {y -1}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {y -1}{y}} \,dy &= \frac {1}{x} \,d x \\ \int { \frac {1}{\frac {y -1}{y}} \,dy} &= \int {\frac {1}{x} \,d x} \\ y +\ln \left (y -1\right )&=\ln \left (x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \operatorname {LambertW}\left (x \,{\mathrm e}^{c_{1} -1}\right )+1 \\ \end{align*} Since \(c_{1}\) is constant, then exponential powers of this constant are constants also, and these can be simplified to just \(c_{1}\) in the above solution. Which simplifies to \[ y = \operatorname {LambertW}\left (x \,{\mathrm e}^{c_{1} -1}\right )+1 \] gives \[ y = \operatorname {LambertW}\left (x c_{1} {\mathrm e}^{-1}\right )+1 \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \operatorname {LambertW}\left (x c_{1} {\mathrm e}^{-1}\right )+1 \\ \end{align*}
Verification of solutions
\[ y = \operatorname {LambertW}\left (x c_{1} {\mathrm e}^{-1}\right )+1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y y^{\prime }-y=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y-1}{x y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } y}{y-1}=\frac {1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } y}{y-1}d x =\int \frac {1}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y+\ln \left (y-1\right )=\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\mathit {LambertW}\left (x \,{\mathrm e}^{c_{1} -1}\right )+1 \end {array} \]
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: 0.016 (sec). Leaf size: 12
dsolve(x*y(x)*diff(y(x),x)=y(x)-1,y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {LambertW}\left (x c_{1} {\mathrm e}^{-1}\right )+1 \]
✓ Solution by Mathematica
Time used: 3.215 (sec). Leaf size: 21
DSolve[x*y[x]*y'[x]==y[x]-1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 1+W\left (e^{-1+c_1} x\right ) \\ y(x)\to 1 \\ \end{align*}