Internal problem ID [12901]
Internal file name [OUTPUT/11554_Tuesday_November_07_2023_11_26_59_PM_59055855/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 1.3 page 47
Problem number: 3.
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]
\[ \boxed {y^{\prime }+2 y=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{1-2 y}d y &= t +c_{1}\\ -\frac {\ln \left (y -\frac {1}{2}\right )}{2}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-2 t -2 c_{1}}+\frac {1}{2}\\ &=\frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}+\frac {1}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}+\frac {1}{2} \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-2 t}}{c_{1}^{2}}+\frac {1}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+2 y=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1-2 y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1-2 y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{1-2 y}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (1-2 y\right )}{2}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {{\mathrm e}^{-2 t -2 c_{1}}}{2}+\frac {1}{2} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 12
dsolve(diff(y(t),t)=1-2*y(t),y(t), singsol=all)
\[ y \left (t \right ) = {\mathrm e}^{-2 t} c_{1} +\frac {1}{2} \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 24
DSolve[y'[t]==1-2*y[t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{2}+c_1 e^{-2 t} \\ y(t)\to \frac {1}{2} \\ \end{align*}