Internal problem ID [13356]
Internal file name [OUTPUT/12528_Wednesday_February_14_2024_11_55_23_PM_35191/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 5. LINEAR FIRST ORDER EQUATIONS. Additional exercises. page
103
Problem number: 5.2 (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]
\[ \boxed {y^{\prime }+2 y=6} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-2 y +6}d y &= x +c_{1}\\ -\frac {\ln \left (y -3\right )}{2}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-2 x -2 c_{1}}+3\\ &=\frac {{\mathrm e}^{-2 x}}{c_{1}^{2}}+3 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-2 x}}{c_{1}^{2}}+3 \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-2 x}}{c_{1}^{2}}+3 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+2 y=6 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-2 y+6 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-2 y+6}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-2 y+6}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-y+3\right )}{2}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-{\mathrm e}^{-2 x -2 c_{1}}+3 \end {array} \]
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)+2*y(x)=6,y(x), singsol=all)
\[ y \left (x \right ) = 3+{\mathrm e}^{-2 x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.03 (sec). Leaf size: 20
DSolve[y'[x]+2*y[x]==6,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 3+c_1 e^{-2 x} \\ y(x)\to 3 \\ \end{align*}