2.2 problem Problem 2

Internal problem ID [2623]
Internal file name [OUTPUT/2115_Sunday_June_05_2022_02_49_05_AM_4677793/index.tex]

Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section: Chapter 1, First-Order Differential Equations. Section 1.4, Separable Differential Equations. page 43
Problem number: Problem 2.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program :

Maple gives the following as the ode type

[_separable]

\[ \boxed {y^{\prime }-\frac {y^{2}}{x^{2}+1}=0} \]

2.2.1 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y^{2}}{x^{2}+1} \end {align*}

Where \(f(x)=\frac {1}{x^{2}+1}\) and \(g(y)=y^{2}\). Integrating both sides gives \begin{align*} \frac {1}{y^{2}} \,dy &= \frac {1}{x^{2}+1} \,d x \\ \int { \frac {1}{y^{2}} \,dy} &= \int {\frac {1}{x^{2}+1} \,d x} \\ -\frac {1}{y}&=\arctan \left (x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= -\frac {1}{\arctan \left (x \right )+c_{1}} \\ \end{align*}

Summary

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

Figure 46: Slope field plot

Verification of solutions

\[ y = -\frac {1}{\arctan \left (x \right )+c_{1}} \] Verified OK.

2.2.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {y^{2}}{x^{2}+1}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2}}{x^{2}+1} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}}=\frac {1}{x^{2}+1} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}}d x =\int \frac {1}{x^{2}+1}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{y}=\arctan \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{\arctan \left (x \right )+c_{1}} \end {array} \]

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: 12

dsolve(diff(y(x),x)=y(x)^2/(x^2+1),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {1}{-\arctan \left (x \right )+c_{1}} \]

✓ Solution by Mathematica

Time used: 0.16 (sec). Leaf size: 19

DSolve[y'[x]==y[x]^2/(x^2+1),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{\arctan (x)+c_1} \\ y(x)\to 0 \\ \end{align*}