2.3 problem Differential equations with Linear Coefficients. Exercise 8.3, page 69

Internal problem ID [4443]
Internal file name [OUTPUT/3936_Sunday_June_05_2022_11_52_07_AM_25632392/index.tex]

Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 8
Problem number: Differential equations with Linear Coefficients. Exercise 8.3, page 69.
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]

\[ \boxed {y+\left (2 x +2 y+2\right ) y^{\prime }=-x -1} \]

2.3.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { -{\frac {1}{2}}\,\mathop {\mathrm {d}x}}\\ &= -\frac {x}{2}+c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x}{2}+c_{1} \\ \end{align*}

Figure 29: Slope field plot

Verification of solutions

\[ y = -\frac {x}{2}+c_{1} \] Verified OK.

2.3.2 Maple step by step solution

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

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 17

dsolve((x+y(x)+1)+(2*x+2*y(x)+2)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -1-x \\ y \left (x \right ) &= -\frac {x}{2}+c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 22

DSolve[(x+y[x]+1)+(2*x+2*y[x]+2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -x-1 \\ y(x)\to -\frac {x}{2}+c_1 \\ \end{align*}