2.15 problem 19

Internal problem ID [25]
Internal file name [OUTPUT/25_Sunday_June_05_2022_01_33_51_AM_10109649/index.tex]

Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.3. Slope fields and solution curves. Page 26
Problem number: 19.
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]

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

2.15.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= \ln \left (y^{2}+1\right ) \end {align*}

The \(y\) domain of \(f(x,y)\) when \(x=0\) is \[ \{-\infty

The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(x=0\) is \[ \{-\infty

2.15.2 Solving as quadrature ode

Since ode has form \(y^{\prime }= f(y)\) and initial conditions \(y = 0\) is verified to satisfy the ode, then the solution is \begin {align*} y&=y_0 \\ &=0 \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 0 \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = 0 \] Verified OK.

2.15.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-\ln \left (1+y^{2}\right )=0, y \left (0\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\ln \left (1+y^{2}\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\ln \left (1+y^{2}\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\ln \left (1+y^{2}\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\ln \left (1+y^{2}\right )}d x =x +c_{1} \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \end {array} \]

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: 0.0 (sec). Leaf size: 5

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

\[ y \left (x \right ) = 0 \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 6

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

\[ y(x)\to 0 \]