Internal problem ID [11985]
Internal file name [OUTPUT/10638_Saturday_September_02_2023_02_48_51_PM_15979896/index.tex]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 8, Separable equations. Exercises page 72
Problem number: 8.1 (iv).
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]
\[ \boxed {x^{\prime }+x^{2}=0} \]
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*}
Verification of solutions
\[ x = \frac {1}{t +c_{1}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }+x^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=-x^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{x^{2}}=-1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{x^{2}}d t =\int \left (-1\right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{x}=-t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=-\frac {1}{-t +c_{1}} \end {array} \]
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: 0.169 (sec). Leaf size: 18
DSolve[x'[t]==-x[t]^2,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{t-c_1} \\ x(t)\to 0 \\ \end{align*}