Internal problem ID [5923]
Internal file name [OUTPUT/5171_Sunday_June_05_2022_03_26_43_PM_81309885/index.tex]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 1.6 Introduction– Linear equations of First Order. Page 41
Problem number: 1(a).
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]
\[ \boxed {-2 y+y^{\prime }=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{1+2 y}d y &= x +c_{1}\\ \frac {\ln \left (\frac {1}{2}+y \right )}{2}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{2 x +2 c_{1}}-\frac {1}{2}\\ &={\mathrm e}^{2 x} c_{1}^{2}-\frac {1}{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{2 x} c_{1}^{2}-\frac {1}{2} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{2 x} c_{1}^{2}-\frac {1}{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -2 y+y^{\prime }=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=1+2 y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{1+2 y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{1+2 y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (1+2 y\right )}{2}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {{\mathrm e}^{2 x +2 c_{1}}}{2}-\frac {1}{2} \end {array} \]
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: 12
dsolve(diff(y(x),x)-2*y(x)=1,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {1}{2}+{\mathrm e}^{2 x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 24
DSolve[y'[x]-2*y[x]==1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2}+c_1 e^{2 x} \\ y(x)\to -\frac {1}{2} \\ \end{align*}