Internal problem ID [542]
Internal file name [OUTPUT/542_Sunday_June_05_2022_01_43_59_AM_906908/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.5. Page 88
Problem number: 13.
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]
\[ \boxed {y^{\prime }-\left (1-y\right )^{2} y^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{2} \left (y -1\right )^{2}}d y &= \int {dt}\\ \int _{}^{y}\frac {1}{\textit {\_a}^{2} \left (\textit {\_a} -1\right )^{2}}d \textit {\_a}&= t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{\textit {\_a}^{2} \left (\textit {\_a} -1\right )^{2}}d \textit {\_a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {1}{\textit {\_a}^{2} \left (\textit {\_a} -1\right )^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\left (1-y\right )^{2} y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (1-y\right )^{2} y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (1-y\right )^{2} y^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{\left (1-y\right )^{2} y^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{-1+y}-2 \ln \left (-1+y\right )-\frac {1}{y}+2 \ln \left (y\right )=t +c_{1} \end {array} \]
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: 0.016 (sec). Leaf size: 66
dsolve(diff(y(t),t) = (1-y(t))^2*y(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (-2 \ln \left ({\mathrm e}^{\textit {\_Z}}+1\right ) {\mathrm e}^{2 \textit {\_Z}}+c_{1} {\mathrm e}^{2 \textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{2 \textit {\_Z}}+t \,{\mathrm e}^{2 \textit {\_Z}}-2 \ln \left ({\mathrm e}^{\textit {\_Z}}+1\right ) {\mathrm e}^{\textit {\_Z}}+c_{1} {\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+t \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}+1\right )}+1 \]
✓ Solution by Mathematica
Time used: 0.365 (sec). Leaf size: 50
DSolve[y'[t] == (1-y[t])^2*y[t]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \text {InverseFunction}\left [-\frac {1}{\text {$\#$1}-1}-\frac {1}{\text {$\#$1}}-2 \log (1-\text {$\#$1})+2 \log (\text {$\#$1})\&\right ][t+c_1] \\ y(t)\to 0 \\ y(t)\to 1 \\ \end{align*}