Internal problem ID [541]
Internal file name [OUTPUT/541_Sunday_June_05_2022_01_43_58_AM_15077121/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.5. Page 88
Problem number: 12.
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]
\[ \boxed {y^{\prime }-y^{2} \left (4-y^{2}\right )=0} \]
Integrating both sides gives \begin {align*} \int -\frac {1}{y^{2} \left (y^{2}-4\right )}d y &= \int {dt}\\ \int _{}^{y}-\frac {1}{\textit {\_a}^{2} \left (\textit {\_a}^{2}-4\right )}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}^{2}-4\right )}d \textit {\_a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {1}{\textit {\_a}^{2} \left (\textit {\_a}^{2}-4\right )}d \textit {\_a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2} \left (4-y^{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2} \left (4-y^{2}\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2} \left (4-y^{2}\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{2} \left (4-y^{2}\right )}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-2+y\right )}{16}-\frac {1}{4 y}+\frac {\ln \left (2+y\right )}{16}=t +c_{1} \end {array} \]
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: 0.032 (sec). Leaf size: 49
dsolve(diff(y(t),t) = y(t)^2*(4-y(t)^2),y(t), singsol=all)
\[ y \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-4\right ) {\mathrm e}^{\textit {\_Z}}+16 c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+16 t \,{\mathrm e}^{\textit {\_Z}}-2 \ln \left ({\mathrm e}^{\textit {\_Z}}-4\right )-32 c_{1} +2 \textit {\_Z} -32 t +4\right )}-2 \]
✓ Solution by Mathematica
Time used: 0.247 (sec). Leaf size: 57
DSolve[y'[t] == y[t]^2*(4-y[t]^2),y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \text {InverseFunction}\left [\frac {1}{4 \text {$\#$1}}+\frac {1}{16} \log (2-\text {$\#$1})-\frac {1}{16} \log (\text {$\#$1}+2)\&\right ][-t+c_1] \\ y(t)\to -2 \\ y(t)\to 0 \\ y(t)\to 2 \\ \end{align*}