Internal problem ID [6142]
Internal file name [OUTPUT/5390_Sunday_June_05_2022_03_36_02_PM_91456469/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 1.3 SEPARABLE EQUATIONS. Page
12
Problem number: 1(d).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+y^{2}=-1} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {-y^{2}-1}{x^{2}+1} \end {align*}
Where \(f(x)=\frac {1}{x^{2}+1}\) and \(g(y)=-y^{2}-1\). Integrating both sides gives \begin{align*} \frac {1}{-y^{2}-1} \,dy &= \frac {1}{x^{2}+1} \,d x \\ \int { \frac {1}{-y^{2}-1} \,dy} &= \int {\frac {1}{x^{2}+1} \,d x} \\ -\arctan \left (y \right )&=\arctan \left (x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= -\tan \left (\arctan \left (x \right )+c_{1} \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= -\tan \left (\arctan \left (x \right )+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = -\tan \left (\arctan \left (x \right )+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}+1\right ) y^{\prime }+y^{2}=-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-1-y^{2}}{x^{2}+1} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-1-y^{2}}=\frac {1}{x^{2}+1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-1-y^{2}}d x =\int \frac {1}{x^{2}+1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\arctan \left (y\right )=\arctan \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\tan \left (\arctan \left (x \right )+c_{1} \right ) \end {array} \]
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: 0.0 (sec). Leaf size: 11
dsolve((1+x^2)*diff(y(x),x)+1+y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = -\tan \left (\arctan \left (x \right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.233 (sec). Leaf size: 29
DSolve[(1+x^2)*y'[x]+1+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\tan (\arctan (x)-c_1) \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}