Consider the inhomogeneous differential equation (d u)/(d t)=a u+b, where a, b are constants. (a

step 1 So here in this question first we find the equilibrium point the equilibrium point an equilibriu

Consider the inhomogeneous differential equation (d u)/(d...

Consider the inhomogeneous differential equation $\frac{d u}{d t}=a u+b$, where $a, b$ are constants. (a) Show that $u_{\star}=-b / a$ is a constant equilibrium solution. (b) Solve the differential equation. Hint: Look at the differential equation satisfied by $v=u-u_*$. (c) Discuss the stability of the equilibrium solution $u_{\star}$ *

Consider the inhomogeneous differential equation $\frac{d u}{d t}=a u+b$, where $a, b$ are constants.
(a) Show that $u_{\star}=-b / a$ is a constant equilibrium solution. (b) Solve the differential equation. Hint: Look at the differential equation satisfied by $v=u-u_*$.
(c) Discuss the stability of the equilibrium solution $u_{\star}$ *

Key Concepts

Equilibrium Solution

An equilibrium solution of a differential equation is a constant solution where the derivative vanishes. For the equation du/dt = au + b, setting the derivative to zero leads to au + b = 0, which gives u* = -b/a. This constant solution represents a steady state where the system remains if it starts there.

Substitution Method

By defining a new variable v = u - u*, where u* is the equilibrium solution, the original inhomogeneous differential equation can be transformed into a homogeneous one. This substitution simplifies the equation to one of the form dv/dt = av, isolating the dynamics of deviations from the equilibrium.

Solving First-Order Linear ODEs

First-order linear ordinary differential equations, such as du/dt = au + b, are typically solved using techniques like the integrating factor method or by reducing the inhomogeneous equation to a homogeneous one via substitution. The general solution obtained usually comprises a transient exponential term that decays or grows over time added to the equilibrium solution.

Stability Analysis

Stability analysis involves examining how small perturbations around an equilibrium behave over time. For the equation in question, once reduced to dv/dt = av, the sign of the constant 'a' determines stability: if a is negative, the exponential term decays, which indicates that u* is a stable equilibrium; if a is positive, the term grows, making u* unstable.

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