Internal problem ID [12870]
Internal file name [OUTPUT/11523_Monday_November_06_2023_01_31_18_PM_88496519/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.2. page
33
Problem number: 10.
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]
\[ \boxed {x^{\prime }-x^{2}=1} \]
Integrating both sides gives \begin {align*} \int \frac {1}{x^{2}+1}d x &= t +c_{1}\\ \arctan \left (x \right )&=t +c_{1} \end {align*}
Solving for \(x\) gives these solutions \begin {align*} x_1&=\tan \left (t +c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= \tan \left (t +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ x = \tan \left (t +c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }-x^{2}=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & x^{\prime }=1+x^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {x^{\prime }}{1+x^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {x^{\prime }}{1+x^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \arctan \left (x\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=\tan \left (t +c_{1} \right ) \end {array} \]
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: 0.016 (sec). Leaf size: 8
dsolve(diff(x(t),t)=1+x(t)^2,x(t), singsol=all)
\[ x \left (t \right ) = \tan \left (t +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.222 (sec). Leaf size: 24
DSolve[x'[t]==1+x[t]^2,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \tan (t+c_1) \\ x(t)\to -i \\ x(t)\to i \\ \end{align*}