Internal problem ID [12639]
Internal file name [OUTPUT/11292_Friday_November_03_2023_06_29_54_AM_80971990/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.2, page 53
Problem number: 4.
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]
\[ \boxed {y^{\prime }-\ln \left (y-1\right )=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{\ln \left (y -1\right )}d y &= \int {dx}\\ \int _{}^{y}\frac {1}{\ln \left (\textit {\_a} -1\right )}d \textit {\_a}&= x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{\ln \left (\textit {\_a} -1\right )}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {1}{\ln \left (\textit {\_a} -1\right )}d \textit {\_a} = x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\ln \left (y-1\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\ln \left (y-1\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\ln \left (y-1\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\ln \left (y-1\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\mathrm {Ei}_{1}\left (-\ln \left (y-1\right )\right )=x +c_{1} \end {array} \]
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: 0.0 (sec). Leaf size: 16
dsolve(diff(y(x),x)=ln(y(x)-1),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (x +\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right )+c_{1} \right )}+1 \]
✓ Solution by Mathematica
Time used: 0.29 (sec). Leaf size: 21
DSolve[y'[x]==Log[y[x]-1],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}[\operatorname {LogIntegral}(\text {$\#$1}-1)\&][x+c_1] \\ y(x)\to 2 \\ \end{align*}