9.36 problem 38 part (a)

Internal problem ID [1142]
Internal file name [OUTPUT/1143_Sunday_June_05_2022_02_03_27_AM_95477654/index.tex]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number: 38 part (a).
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]

\[ \boxed {y^{\prime }+y^{2}=-k^{2}} \]

9.36.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int \frac {1}{-k^{2}-y^{2}}d y &= x +c_{1}\\ -\frac {\arctan \left (\frac {y}{k}\right )}{k}&=x +c_{1} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} y_1&=-\tan \left (c_{1} k +k x \right ) k \end {align*}

Summary

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

Verification of solutions

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

9.36.2 Maple step by step solution

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

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: 0.015 (sec). Leaf size: 13

dsolve(diff(y(x),x)+y(x)^2+k^2=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 4.112 (sec). Leaf size: 35

DSolve[y'[x]+y[x]^2+k^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -k \tan (k (x-c_1)) \\ y(x)\to -i k \\ y(x)\to i k \\ \end{align*}