Internal problem ID [12617]
Internal file name [OUTPUT/11270_Friday_November_03_2023_06_29_33_AM_11055370/index.tex]
Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.1, page 40
Problem number: 2 (D).
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]
\[ \boxed {y^{\prime }-y=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{1+y}d y &= x +c_{1}\\ \ln \left (1+y \right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{x +c_{1}}-1\\ &={\mathrm e}^{x} c_{1} -1 \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{x} c_{1} -1 \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{x} c_{1} -1 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1+y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1+y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{1+y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (1+y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{x +c_{1}}-1 \end {array} \]
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: 10
dsolve(diff(y(x),x)=1+y(x),y(x), singsol=all)
\[ y \left (x \right ) = -1+c_{1} {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.037 (sec). Leaf size: 18
DSolve[y'[x]==1+y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -1+c_1 e^x \\ y(x)\to -1 \\ \end{align*}