3.7 problem 6

Internal problem ID [11365]
Internal file name [OUTPUT/10348_Wednesday_May_17_2023_07_49_41_PM_7363715/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 1.2 Antiderivatives. Exercises page 19
Problem number: 6.
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]

\[ \boxed {x^{\prime }=\frac {{\mathrm e}^{-t}}{\sqrt {t}}} \] With initial conditions \begin {align*} [x \left (1\right ) = 0] \end {align*}

3.7.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} x^{\prime } + p(t)x &= q(t) \end {align*}

Where here \begin {align*} p(t) &=0\\ q(t) &=\frac {{\mathrm e}^{-t}}{\sqrt {t}} \end {align*}

Hence the ode is \begin {align*} x^{\prime } = \frac {{\mathrm e}^{-t}}{\sqrt {t}} \end {align*}

The domain of \(p(t)=0\) is \[ \{-\infty

3.7.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} x &= \int { \frac {{\mathrm e}^{-t}}{\sqrt {t}}\,\mathop {\mathrm {d}t}}\\ &= \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {t}\right )+c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(t=1\) and \(x=0\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 0 = \operatorname {erf}\left (1\right ) \sqrt {\pi }+c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = -\operatorname {erf}\left (1\right ) \sqrt {\pi } \end {align*}

Trying the constant \begin {align*} c_{1} = -\operatorname {erf}\left (1\right ) \sqrt {\pi } \end {align*}

Substituting this in the general solution gives \begin {align*} x&=\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {t}\right )-\operatorname {erf}\left (1\right ) \sqrt {\pi } \end {align*}

The constant \(c_{1} = -\operatorname {erf}\left (1\right ) \sqrt {\pi }\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {t}\right )-\operatorname {erf}\left (1\right ) \sqrt {\pi } \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ x = \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {t}\right )-\operatorname {erf}\left (1\right ) \sqrt {\pi } \] Verified OK.

3.7.3 Maple step by step solution

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

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful`
 

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 17

dsolve([diff(x(t),t)=exp(-t)/sqrt(t),x(1) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \left (-\operatorname {erf}\left (1\right )+\operatorname {erf}\left (\sqrt {t}\right )\right ) \sqrt {\pi } \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 22

DSolve[{x'[t]==Exp[-t]/Sqrt[t],{x[1]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \sqrt {\pi } \left (\text {erf}\left (\sqrt {t}\right )-\text {erf}(1)\right ) \]