Internal problem ID [12969]
Internal file name [OUTPUT/11622_Tuesday_November_07_2023_11_52_13_PM_76106352/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: 7.
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]
\[ \boxed {v^{\prime }+v^{2}+2 v=-2} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-v^{2}-2 v -2}d v &= t +c_{1}\\ -\arctan \left (v +1\right )&=t +c_{1} \end {align*}
Solving for \(v\) gives these solutions \begin {align*} v_1&=-1-\tan \left (t +c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} v &= -1-\tan \left (t +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ v = -1-\tan \left (t +c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & v^{\prime }+v^{2}+2 v=-2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & v^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & v^{\prime }=-v^{2}-2 v-2 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {v^{\prime }}{-v^{2}-2 v-2}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {v^{\prime }}{-v^{2}-2 v-2}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\arctan \left (v+1\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \\ {} & {} & v=-1-\tan \left (t +c_{1} \right ) \end {array} \]
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: 12
dsolve(diff(v(t),t)=-v(t)^2-2*v(t)-2,v(t), singsol=all)
\[ v \left (t \right ) = -1-\tan \left (t +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.699 (sec). Leaf size: 30
DSolve[v'[t]==-v[t]^2-2*v[t]-2,v[t],t,IncludeSingularSolutions -> True]
\begin{align*} v(t)\to -1-\tan (t-c_1) \\ v(t)\to -1-i \\ v(t)\to -1+i \\ \end{align*}