Question

Reread the discussion following example 5 and construct a linear first-order differential equation for which all solutions are asymptotic to the line.

$y=3x-5 as x\to \infty$

Short answer

The differential equation$y\text{'}+y=3x-2$ hasasymptotic to the line$y=3x-5$

Step-by-step solution

Definition of a Differential Equation

Differential Equation is defined as the equation that consists of the derivatives of one or more dependent functions with respect to one or more independent functions.

$dy/dx=f\left(x\right)$

Determination of a differential equation

First order linear differential equation

$y\text{'}+P\left(x\right)y=f\left(x\right)$

The general equation;

$y=3x-5+c{e}^{-x}$

Differentiating the above equation,

$$\begin{gathered} y\text{'}=3-c{e}^{-x} \\ =3-(y-3x+5) \\ =3-y+3x-5 \\ =3x-y-2 \end{gathered}$$

Hence, the differential equation $y\text{'}+y=3x-2$hasasymptotic to the line $y=3x-5$