1.8 problem 22

Internal problem ID [12580]
Internal file name [OUTPUT/11233_Thursday_October_19_2023_04_43_32_PM_53349013/index.tex]

Book: Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section: Chapter 1. Introduction. Exercises page 14
Problem number: 22.
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]

\[ \boxed {y^{\prime }-2 \sqrt {{| y|}}=0} \]

1.8.1 Solving as quadrature ode

Integrating both sides gives \begin{align*} \int \frac {1}{2 \sqrt {{| y |}}}d y &= \int d x \\ \frac {\left (\left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 0

Summary

The solution(s) found are the following \begin{align*} \tag{1} \frac {\left (\left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 0

Figure 13: Slope field plot

Verification of solutions

\[ \frac {\left (\left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 0

1.8.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-2 \sqrt {{| 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 }=2 \sqrt {{| y|}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {{| y|}}}=2 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {{| y|}}}d x =\int 2d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 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: 0.015 (sec). Leaf size: 27

dsolve(diff(y(x),x)-2*sqrt(abs(y(x)))=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.291 (sec). Leaf size: 31

DSolve[y'[x]-Sqrt[Abs[y[x]]]==0,y[x],x,IncludeSingularSolutions -> True]
 

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