3.9 problem 8.5

Internal problem ID [11990]
Internal file name [OUTPUT/10643_Saturday_September_02_2023_02_48_54_PM_64009715/index.tex]

Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section: Chapter 8, Separable equations. Exercises page 72
Problem number: 8.5.
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]

\[ \boxed {x^{\prime }-\lambda x=0} \]

3.9.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {1}{\lambda x}d x &= t +c_{1}\\ \frac {\ln \left (x \right )}{\lambda }&=t +c_{1} \end {align*}

Solving for \(x\) gives these solutions \begin {align*} x_1&={\mathrm e}^{c_{1} \lambda +t \lambda }\\ &=c_{1} {\mathrm e}^{t \lambda } \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= c_{1} {\mathrm e}^{t \lambda } \\ \end{align*}

Verification of solutions

\[ x = c_{1} {\mathrm e}^{t \lambda } \] Verified OK.

3.9.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-\lambda x=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=\lambda x \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{x}=\lambda \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{x}d t =\int \lambda d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (x\right )=t \lambda +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x={\mathrm e}^{t \lambda +c_{1}} \end {array} \]

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(x(t),t)=lambda*x(t),x(t), singsol=all)
 

\[ x \left (t \right ) = c_{1} {\mathrm e}^{\lambda t} \]

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 18

DSolve[x'[t]==\[Lambda]*x[t],x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 e^{\lambda t} \\ x(t)\to 0 \\ \end{align*}