Internal problem ID [5920]
Internal file name [OUTPUT/5168_Sunday_June_05_2022_03_26_40_PM_97624927/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 1.3 Introduction– Linear equations of First Order. Page 38
Problem number: 3(a).
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]
\[ \boxed {y^{\prime }+5 y=2} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-5 y +2}d y &= x +c_{1}\\ -\frac {\ln \left (y -\frac {2}{5}\right )}{5}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-5 x -5 c_{1}}+\frac {2}{5}\\ &=\frac {{\mathrm e}^{-5 x}}{c_{1}^{5}}+\frac {2}{5} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-5 x}}{c_{1}^{5}}+\frac {2}{5} \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-5 x}}{c_{1}^{5}}+\frac {2}{5} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+5 y=2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-5 y+2 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-5 y+2}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-5 y+2}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-5 y+2\right )}{5}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {{\mathrm e}^{-5 x -5 c_{1}}}{5}+\frac {2}{5} \end {array} \]
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(x),x)+5*y(x)=2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {2}{5}+{\mathrm e}^{-5 x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 24
DSolve[y'[x]+5*y[x]==2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2}{5}+c_1 e^{-5 x} \\ y(x)\to \frac {2}{5} \\ \end{align*}