Internal problem ID [13302]
Internal file name [OUTPUT/12474_Wednesday_February_14_2024_02_06_35_AM_5860910/index.tex
]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page
90
Problem number: 4.3 (e).
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]
\[ \boxed {y^{\prime }+4 y=8} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-4 y +8}d y &= x +c_{1}\\ -\frac {\ln \left (y -2\right )}{4}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-4 x -4 c_{1}}+2\\ &=\frac {{\mathrm e}^{-4 x}}{c_{1}^{4}}+2 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-4 x}}{c_{1}^{4}}+2 \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-4 x}}{c_{1}^{4}}+2 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+4 y=8 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-4 y+8 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-4 y+8}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-4 y+8}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-y+2\right )}{4}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-{\mathrm e}^{-4 x -4 c_{1}}+2 \end {array} \]
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(x),x)+4*y(x)=8,y(x), singsol=all)
\[ y \left (x \right ) = 2+{\mathrm e}^{-4 x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 20
DSolve[y'[x]+4*y[x]==8,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2+c_1 e^{-4 x} \\ y(x)\to 2 \\ \end{align*}