1.3 problem 3

Internal problem ID [11350]
Internal file name [OUTPUT/10333_Wednesday_May_17_2023_07_49_20_PM_54653337/index.tex]

Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section: Chapter 1, First order differential equations. Section 1.1 First order equations. Exercises page 10
Problem number: 3.
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]

\[ \boxed {x^{\prime }+x^{2}=0} \]

1.3.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int -\frac {1}{x^{2}}d x &= t +c_{1}\\ \frac {1}{x}&=t +c_{1} \end {align*}

Solving for \(x\) gives these solutions \begin {align*} x_1&=\frac {1}{t +c_{1}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= \frac {1}{t +c_{1}} \\ \end{align*}

Figure 11: Slope field plot

Verification of solutions

\[ x = \frac {1}{t +c_{1}} \] Verified OK.

1.3.2 Maple step by step solution

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

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
<- Bernoulli successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 9

dsolve(diff(x(t),t)=-x(t)^2,x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {1}{t +c_{1}} \]

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 39

DSolve[x'[t]==-t/x[t],x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\sqrt {-t^2+2 c_1} \\ x(t)\to \sqrt {-t^2+2 c_1} \\ \end{align*}