Internal problem ID [6123]
Internal file name [OUTPUT/5371_Sunday_June_05_2022_03_35_37_PM_74157158/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS.
Page 9
Problem number: 2(e).
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]
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }=\arctan \left (x \right )} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {\arctan \left (x \right )}{x^{2}+1}\,\mathop {\mathrm {d}x}}\\ &= \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}+1\right ) y^{\prime }=\arctan \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\arctan \left (x \right )}{x^{2}+1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\arctan \left (x \right )}{x^{2}+1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\arctan \left (x \right )^{2}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\arctan \left (x \right )^{2}}{2}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 12
dsolve((1+x^2)*diff(y(x),x)=arctan(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\arctan \left (x \right )^{2}}{2}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 16
DSolve[(1+x^2)*y'[x]==ArcTan[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\arctan (x)^2}{2}+c_1 \]