Internal problem ID [13351]
Internal file name [OUTPUT/12523_Wednesday_February_14_2024_11_54_40_PM_41001360/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: 5.1 (f).
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\\ &={\mathrm e}^{4 x} c_{1}^{4}-2 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{4 x} c_{1}^{4}-2 \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{4 x} c_{1}^{4}-2 \] 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: 0.015 (sec). Leaf size: 12
dsolve(diff(y(x),x)=4*y(x)+8,y(x), singsol=all)
\[ y \left (x \right ) = -2+c_{1} {\mathrm e}^{4 x} \]
✓ Solution by Mathematica
Time used: 0.032 (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*}