Internal problem ID [13259]
Internal file name [OUTPUT/12431_Wednesday_February_14_2024_02_06_12_AM_11754521/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page
32
Problem number: 2.3 (h).
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]
\[ \boxed {-\left (x^{2}-9\right ) y^{\prime }=-1} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {1}{x^{2}-9}\,\mathop {\mathrm {d}x}}\\ &= -\frac {\operatorname {arctanh}\left (\frac {x}{3}\right )}{3}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\operatorname {arctanh}\left (\frac {x}{3}\right )}{3}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\operatorname {arctanh}\left (\frac {x}{3}\right )}{3}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\left (x^{2}-9\right ) y^{\prime }=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{x^{2}-9} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{x^{2}-9}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\ln \left (x -3\right )}{6}-\frac {\ln \left (3+x \right )}{6}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\ln \left (x -3\right )}{6}-\frac {\ln \left (3+x \right )}{6}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 18
dsolve(1=(x^2-9)*diff(y(x),x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\ln \left (-3+x \right )}{6}-\frac {\ln \left (x +3\right )}{6}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 26
DSolve[1==(x^2-9)*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{6} (\log (3-x)-\log (x+3)+6 c_1) \]