3.18 problem 13

Internal problem ID [12631]
Internal file name [OUTPUT/11284_Friday_November_03_2023_06_29_41_AM_62814595/index.tex]

Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 2. The Initial Value Problem. Exercises 2.1, page 40
Problem number: 13.
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]

\[ \boxed {y^{\prime }-{| y|}=0} \]

3.18.1 Solving as quadrature ode

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

Solving for \(y\) gives these solutions \begin {align*} y_1&={\mathrm e}^{-x -c_{1}}\\ &=\frac {{\mathrm e}^{-x}}{c_{1}}\\ y_2&={\mathrm e}^{x +c_{1}}\\ &=c_{1} {\mathrm e}^{x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {{\mathrm e}^{-x}}{c_{1}} \\ \tag{2} y &= c_{1} {\mathrm e}^{x} \\ \end{align*}

Figure 124: Slope field plot

Verification of solutions

\[ y = \frac {{\mathrm e}^{-x}}{c_{1}} \] Verified OK.

\[ y = c_{1} {\mathrm e}^{x} \] Verified OK.

3.18.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-{| y|}=0 \\ \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|} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{{| y|}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{{| y|}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \left \{\begin {array}{cc} -\ln \left (y\right ) & y<0 \\ \mathit {undefined} & y=0 \\ \ln \left (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: 0.125 (sec). Leaf size: 19

dsolve(diff(y(x),x)=abs(y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.229 (sec). Leaf size: 29

DSolve[y'[x]==Abs[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{| K[1]| }dK[1]\&\right ][x+c_1] \\ y(x)\to 0 \\ \end{align*}