Internal problem ID [6409]
Internal file name [OUTPUT/5657_Sunday_June_05_2022_03_46_12_PM_90120699/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(c) 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]
\[ \boxed {y^{\prime }-y=2} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y +2}d y &= x +c_{1}\\ \ln \left (y +2\right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{x +c_{1}}-2\\ &=c_{1} {\mathrm e}^{x}-2 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{x}-2 \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{x}-2 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y=2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=2+y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{2+y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{2+y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (2+y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{x +c_{1}}-2 \end {array} \]
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: 10
dsolve(diff(y(x),x)-y(x)=2,y(x), singsol=all)
\[ y \left (x \right ) = -2+{\mathrm e}^{x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 18
DSolve[y'[x]-y[x]==2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -2+c_1 e^x \\ y(x)\to -2 \\ \end{align*}