Internal problem ID [898]
Internal file name [OUTPUT/898_Sunday_June_05_2022_01_53_32_AM_13100521/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
Problem number: 12.
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]
\[ \boxed {3 y+y^{\prime }=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-3 y +1}d y &= x +c_{1}\\ -\frac {\ln \left (y -\frac {1}{3}\right )}{3}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-3 x -3 c_{1}}+\frac {1}{3}\\ &=\frac {{\mathrm e}^{-3 x}}{c_{1}^{3}}+\frac {1}{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-3 x}}{c_{1}^{3}}+\frac {1}{3} \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-3 x}}{c_{1}^{3}}+\frac {1}{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 y+y^{\prime }=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-3 y+1 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-3 y+1}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-3 y+1}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-3 y+1\right )}{3}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {{\mathrm e}^{-3 x -3 c_{1}}}{3}+\frac {1}{3} \end {array} \]
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) +3*y(x)=1,y(x), singsol=all)
\[ y \left (x \right ) = \frac {1}{3}+c_{1} {\mathrm e}^{-3 x} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 24
DSolve[y'[x] +3*y[x]==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3}+c_1 e^{-3 x} \\ y(x)\to \frac {1}{3} \\ \end{align*}