1.17 problem 9

Internal problem ID [885]
Internal file name [OUTPUT/885_Sunday_June_05_2022_01_53_12_AM_32529587/index.tex]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 1, Introduction. Section 1.2 Page 14
Problem number: 9.
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]

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

1.17.1 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)\\ &= {| y |}+1 \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 \[ \{y <0\boldsymbol {\lor }0

1.17.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {1}{{| y |}+1}d y &= \int {dx}\\ \left \{\begin {array}{cc} -\ln \left (1-y \right ) & y \le 0 \\ \ln \left (y +1\right ) & 0

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

The solutions are \begin {align*} c_{1} = 0 \end {align*}

Trying the constant \begin {align*} c_{1} = 0 \end {align*}

Substituting \(c_{1}\) found above in the general solution gives \begin {align*} \left \{\begin {array}{cc} -\ln \left (1-y \right ) & y \le 0 \\ \ln \left (y +1\right ) & 0

The constant \(c_{1} = 0\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} \left \{\begin {array}{cc} -\ln \left (1-y\right ) & y\le 0 \\ \ln \left (1+y\right ) & 0

Verification of solutions

\[ \left \{\begin {array}{cc} -\ln \left (1-y\right ) & y\le 0 \\ \ln \left (1+y\right ) & 0

1.17.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-{| y|}=1, 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 }={| y|}+1 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{{| y|}+1}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{{| y|}+1}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \left \{\begin {array}{cc} -\ln \left (1-y\right ) & y\le 0 \\ \ln \left (1+y\right ) & 0

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: 0.172 (sec). Leaf size: 19

dsolve([diff(y(x),x) = abs(y(x))+1,y(0) = 0],y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{x}-1 \\ y \left (x \right ) &= 1-{\mathrm e}^{-x} \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

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

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