1.39 problem 2.7 c

Internal problem ID [13280]
Internal file name [OUTPUT/12452_Wednesday_February_14_2024_02_06_20_AM_53756952/index.tex]

Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page 32
Problem number: 2.7 c.
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]

\[ \boxed {y^{\prime }=\frac {1}{x^{2}+1}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}

1.39.1 Existence and uniqueness analysis

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

Where here \begin {align*} p(x) &=0\\ q(x) &=\frac {1}{x^{2}+1} \end {align*}

Hence the ode is \begin {align*} y^{\prime } = \frac {1}{x^{2}+1} \end {align*}

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

1.39.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \frac {1}{x^{2}+1}\,\mathop {\mathrm {d}x}}\\ &= \arctan \left (x \right )+c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=1\) and \(y=0\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 0 = \frac {\pi }{4}+c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = -\frac {\pi }{4} \end {align*}

Trying the constant \begin {align*} c_{1} = -\frac {\pi }{4} \end {align*}

Substituting this in the general solution gives \begin {align*} y&=\arctan \left (x \right )-\frac {\pi }{4} \end {align*}

The constant \(c_{1} = -\frac {\pi }{4}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \arctan \left (x \right )-\frac {\pi }{4} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \arctan \left (x \right )-\frac {\pi }{4} \] Verified OK.

1.39.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {1}{x^{2}+1}, y \left (1\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{x^{2}+1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\arctan \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arctan \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (1\right )=0 \\ {} & {} & 0=\frac {\pi }{4}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =-\frac {\pi }{4} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =-\frac {\pi }{4}\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\arctan \left (x \right )-\frac {\pi }{4} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\arctan \left (x \right )-\frac {\pi }{4} \end {array} \]

Maple trace

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 10

dsolve([diff(y(x),x)=1/(x^2+1),y(1) = 0],y(x), singsol=all)
 

\[ y \left (x \right ) = \arctan \left (x \right )-\frac {\pi }{4} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 13

DSolve[{y'[x]==1/(x^2+1),{y[1]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \arctan (x)-\frac {\pi }{4} \]