Internal problem ID [11968]
Internal file name [OUTPUT/10621_Saturday_September_02_2023_02_48_35_PM_99659885/index.tex]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 5, Trivial differential equations. Exercises page 33
Problem number: 5.1 (ii).
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]
\[ \boxed {y^{\prime }=\frac {1}{x^{2}-1}} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {1}{x^{2}-1}\,\mathop {\mathrm {d}x}}\\ &= -\operatorname {arctanh}\left (x \right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\operatorname {arctanh}\left (x \right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\operatorname {arctanh}\left (x \right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{x^{2}-1} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{x^{2}-1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\mathrm {arctanh}\left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\mathrm {arctanh}\left (x \right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 10
dsolve(diff(y(x),x)=1/(x^2-1),y(x), singsol=all)
\[ y \left (x \right ) = -\operatorname {arctanh}\left (x \right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 26
DSolve[y'[x]==1/(x^2-1),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} (\log (1-x)-\log (x+1)+2 c_1) \]