Internal problem ID [12973]
Internal file name [OUTPUT/11626_Tuesday_November_07_2023_11_52_16_PM_91685053/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.6 page 89
Problem number: 11.
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]
\[ \boxed {y^{\prime }-y \ln \left ({| y|}\right )=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y \ln \left ({| y |}\right )}d y &= t +c_{1}\\ \left \{\begin {array}{cc} \ln \left (-\ln \left (-y \right )\right ) & y <0 \\ \operatorname {undefined} & y =0 \\ \ln \left (-\ln \left (y \right )\right ) & 0 Solving for \(y\) gives these solutions \begin {align*} \end {align*}
Summary
The solution(s) found are the following \begin{align*}
\tag{1} y &= {\mathrm e}^{-{\mathrm e}^{t +c_{1}}} \\
\tag{2} y &= -{\mathrm e}^{-{\mathrm e}^{t +c_{1}}} \\
\end{align*} Verification of solutions
\[
y = {\mathrm e}^{-{\mathrm e}^{t +c_{1}}}
\] Verified OK.
\[
y = -{\mathrm e}^{-{\mathrm e}^{t +c_{1}}}
\] Verified OK. \[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y \ln \left ({| y|}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y \ln \left ({| y|}\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y \ln \left ({| y|}\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y \ln \left ({| y|}\right )}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{y \ln \left ({| y|}\right )}d t =t +c_{1} \end {array} \]
Maple trace
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 21
\begin{align*}
y \left (t \right ) &= {\mathrm e}^{-c_{1} {\mathrm e}^{t}} \\
y \left (t \right ) &= -{\mathrm e}^{-c_{1} {\mathrm e}^{t}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.321 (sec). Leaf size: 35
\begin{align*}
y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \log (| K[1]| )}dK[1]\&\right ][t+c_1] \\
y(t)\to 1 \\
\end{align*}
5.24.2 Maple step by step solution
`Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
<- separable successful`
dsolve(diff(y(t),t)=y(t)*ln(abs(y(t))),y(t), singsol=all)
DSolve[y'[t]==y[t]*Log[Abs[y[t]]],y[t],t,IncludeSingularSolutions -> True]