Internal problem ID [12970]
Internal file name [OUTPUT/11623_Tuesday_November_07_2023_11_52_14_PM_25586871/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: 8.
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]
\[ \boxed {w^{\prime }-3 w^{3}+12 w^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{3 w^{3}-12 w^{2}}d w &= \int {dt}\\ \int _{}^{w}\frac {1}{3 \textit {\_a}^{3}-12 \textit {\_a}^{2}}d \textit {\_a}&= t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{w}\frac {1}{3 \textit {\_a}^{3}-12 \textit {\_a}^{2}}d \textit {\_a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{w}\frac {1}{3 \textit {\_a}^{3}-12 \textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & w^{\prime }-3 w^{3}+12 w^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & w^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & w^{\prime }=3 w^{3}-12 w^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {w^{\prime }}{3 w^{3}-12 w^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {w^{\prime }}{3 w^{3}-12 w^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (w-4\right )}{48}+\frac {1}{12 w}-\frac {\ln \left (w\right )}{48}=t +c_{1} \end {array} \]
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: 0.094 (sec). Leaf size: 49
dsolve(diff(w(t),t)=3*w(t)^3-12*w(t)^2,w(t), singsol=all)
\[ w \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}+4\right ) {\mathrm e}^{\textit {\_Z}}+48 c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+48 t \,{\mathrm e}^{\textit {\_Z}}+4 \ln \left ({\mathrm e}^{\textit {\_Z}}+4\right )+192 c_{1} -4 \textit {\_Z} +192 t -4\right )}+4 \]
✓ Solution by Mathematica
Time used: 0.392 (sec). Leaf size: 50
DSolve[w'[t]==3*w[t]^3-12*w[t]^2,w[t],t,IncludeSingularSolutions -> True]
\begin{align*} w(t)\to \text {InverseFunction}\left [\frac {1}{4 \text {$\#$1}}+\frac {1}{16} \log (4-\text {$\#$1})-\frac {\log (\text {$\#$1})}{16}\&\right ][3 t+c_1] \\ w(t)\to 0 \\ w(t)\to 4 \\ \end{align*}