Internal problem ID [6407]
Internal file name [OUTPUT/5655_Sunday_June_05_2022_03_46_09_PM_61613456/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.2. Series Solutions of
First-Order Differential Equations Page 162
Problem number: 1(b) solving directly.
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]
\[ \boxed {y+y^{\prime }=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{1-y}d y &= x +c_{1}\\ -\ln \left (y -1\right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-c_{1} -x}+1\\ &=\frac {{\mathrm e}^{-x}}{c_{1}}+1 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-x}}{c_{1}}+1 \\ \end{align*}
Verification of solutions
\[ y = \frac {{\mathrm e}^{-x}}{c_{1}}+1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y+y^{\prime }=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1-y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1-y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{1-y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\ln \left (1-y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-{\mathrm e}^{-c_{1} -x}+1 \end {array} \]
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)+y(x)=1,y(x), singsol=all)
\[ y \left (x \right ) = 1+c_{1} {\mathrm e}^{-x} \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 20
DSolve[y'[x]+y[x]==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 1+c_1 e^{-x} \\ y(x)\to 1 \\ \end{align*}