Internal problem ID [13253]
Internal file name [OUTPUT/12425_Wednesday_February_14_2024_02_06_11_AM_63328298/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page
32
Problem number: 2.3 (b).
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]
\[ \boxed {y^{\prime }=20 \,{\mathrm e}^{-4 x}} \]
Integrating both sides gives \begin {align*} y &= \int { 20 \,{\mathrm e}^{-4 x}\,\mathop {\mathrm {d}x}}\\ &= -5 \,{\mathrm e}^{-4 x}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -5 \,{\mathrm e}^{-4 x}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -5 \,{\mathrm e}^{-4 x}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=20 \,{\mathrm e}^{-4 x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 20 \,{\mathrm e}^{-4 x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-5 \,{\mathrm e}^{-4 x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-5 \,{\mathrm e}^{-4 x}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 12
dsolve(diff(y(x),x)=20*exp(-4*x),y(x), singsol=all)
\[ y \left (x \right ) = -5 \,{\mathrm e}^{-4 x}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 15
DSolve[y'[x]==20*Exp[-4*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -5 e^{-4 x}+c_1 \]