Internal problem ID [2466]
Internal file name [OUTPUT/1958_Sunday_June_05_2022_02_40_58_AM_15291467/index.tex]
Book: Ordinary Differential Equations, Robert H. Martin, 1983
Section: Problem 1.1-6, page 7
Problem number: 1.1-6 (d).
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]
\[ \boxed {y^{\prime }+y^{2}=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}+1}d y &= t +c_{1}\\ \operatorname {arctanh}\left (y \right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\tanh \left (t +c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \tanh \left (t +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \tanh \left (t +c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1-y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1-y^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{1-y^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \mathrm {arctanh}\left (y\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\tanh \left (t +c_{1} \right ) \end {array} \]
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: 0.0 (sec). Leaf size: 8
dsolve(diff(y(t),t)=1-y(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = \tanh \left (t +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.713 (sec). Leaf size: 44
DSolve[y'[t]==1-y[t]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {e^{2 t}-e^{2 c_1}}{e^{2 t}+e^{2 c_1}} \\ y(t)\to -1 \\ y(t)\to 1 \\ \end{align*}