Internal problem ID [12875]
Internal file name [OUTPUT/11528_Monday_November_06_2023_01_31_21_PM_43900639/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 1.2. page
33
Problem number: 15.
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]
\[ \boxed {y^{\prime }-\frac {1}{2 y+1}=0} \]
Integrating both sides gives \begin {align*} \int \left (2 y +1\right )d y &= t +c_{1}\\ y \left (1+y \right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {1}{2}-\frac {\sqrt {1+4 t +4 c_{1}}}{2}\\ y_2&=-\frac {1}{2}+\frac {\sqrt {1+4 t +4 c_{1}}}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{2}-\frac {\sqrt {1+4 t +4 c_{1}}}{2} \\ \tag{2} y &= -\frac {1}{2}+\frac {\sqrt {1+4 t +4 c_{1}}}{2} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{2}-\frac {\sqrt {1+4 t +4 c_{1}}}{2} \] Verified OK.
\[ y = -\frac {1}{2}+\frac {\sqrt {1+4 t +4 c_{1}}}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {1}{2 y+1}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{2 y+1} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \left (2 y+1\right ) y^{\prime }=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (2 y+1\right ) y^{\prime }d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y^{2}+y=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=-\frac {1}{2}-\frac {\sqrt {1+4 t +4 c_{1}}}{2}, y=-\frac {1}{2}+\frac {\sqrt {1+4 t +4 c_{1}}}{2}\right \} \end {array} \]
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: 0.016 (sec). Leaf size: 35
dsolve(diff(y(t),t)=1/(2*y(t)+1),y(t), singsol=all)
\begin{align*} y \left (t \right ) &= -\frac {1}{2}-\frac {\sqrt {1+4 c_{1} +4 t}}{2} \\ y \left (t \right ) &= -\frac {1}{2}+\frac {\sqrt {1+4 c_{1} +4 t}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.14 (sec). Leaf size: 49
DSolve[y'[t]==1/(2*y[t]+1),y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{2} \left (-1-\sqrt {4 t+1+4 c_1}\right ) \\ y(t)\to \frac {1}{2} \left (-1+\sqrt {4 t+1+4 c_1}\right ) \\ \end{align*}