Internal problem ID [869]
Internal file name [OUTPUT/869_Sunday_June_05_2022_01_52_52_AM_84289303/index.tex
]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 1, Introduction. Section 1.2 Page 14
Problem number: 2(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]
\[ \boxed {y^{\prime }-2 y=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{2 y}d y &= x +c_{1}\\ \frac {\ln \left (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}}\\ &={\mathrm e}^{2 x} c_{1}^{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{2 x} c_{1}^{2} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{2 x} c_{1}^{2} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-2 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=2 y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y}=2 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y}d x =\int 2d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )=2 x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{2 x +c_{1}} \end {array} \]
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) = 2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{2 x} c_{1} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 16
DSolve[y'[x]== y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^x \\ y(x)\to 0 \\ \end{align*}