4.7 problem 1(g)

Internal problem ID [11373]
Internal file name [OUTPUT/10356_Wednesday_May_17_2023_07_49_52_PM_772028/index.tex]

Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section: Chapter 1, First order differential equations. Section 1.3.1 Separable equations. Exercises page 26
Problem number: 1(g).
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]

\[ \boxed {x^{\prime }-{\mathrm e}^{x^{2}}=0} \]

4.7.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int {\mathrm e}^{-x^{2}}d x &= \int {dt}\\ \int _{}^{x}{\mathrm e}^{-\textit {\_a}^{2}}d \textit {\_a}&= t +c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{x}{\mathrm e}^{-\textit {\_a}^{2}}d \textit {\_a} &= t +c_{1} \\ \end{align*}

Figure 39: Slope field plot

Verification of solutions

\[ \int _{}^{x}{\mathrm e}^{-\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.

4.7.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-{\mathrm e}^{x^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }={\mathrm e}^{x^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{{\mathrm e}^{x^{2}}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{{\mathrm e}^{x^{2}}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\sqrt {\pi }\, \mathrm {erf}\left (x\right )}{2}=t +c_{1} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 14

dsolve(diff(x(t),t)=exp(x(t)^2),x(t), singsol=all)
 

\[ t -\frac {\sqrt {\pi }\, \operatorname {erf}\left (x \left (t \right )\right )}{2}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.594 (sec). Leaf size: 17

DSolve[x'[t]==Exp[x[t]^2],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \text {erf}^{-1}\left (\frac {2 (t+c_1)}{\sqrt {\pi }}\right ) \]