(a) Prove that if u1(t) and u2(t) are any two distinct solutions to (d u)/(d t)=a u with a>0. th

step 1 Okay, so we want to go ahead and prove that that if we have a linear differential equation, syst

(a) Prove that if u1(t) and u2(t) are any two distinct...

(a) Prove that if $u_1(t)$ and $u_2(t)$ are any two distinct solutions to $\frac{d u}{d t}=a u$ with $a>0$. then $\left|u_1(t)-u_2(t)\right| \rightarrow \infty$ as $t \rightarrow \infty$. (b) If $a=.02$ and $u_1(0)=.1, u_2(0)=.05$, how long do you have to wait until $\left|u_1(t)-u_2(t)\right|>1,000$ ?

Key Concepts

Exponential Differential Equations

Exponential differential equations, such as those in the form du/dt = a * u, are a fundamental class of ordinary differential equations. They arise when the rate of change of a quantity is directly proportional to its current value. This proportionality leads to solutions that are exponential functions, which serve as essential models for growth and decay processes in a broad range of scientific and engineering contexts.

Separation of Variables

Separation of variables is a method used to solve differential equations by isolating the dependent variable on one side of the equation and the independent variable on the other. This technique is particularly useful for exponential differential equations because it allows one to integrate both sides independently and obtain the general solution in a clear, step-by-step manner.

Initial Value Problems (IVP)

An initial value problem (IVP) involves a differential equation paired with a specific initial condition that the solution must satisfy. This additional piece of information ensures the uniqueness of the solution among all possible solutions of the differential equation, as it determines the constant that appears in the general solution of the exponential equation.

Long-Term Behavior and Divergence of Solutions

The long-term behavior of solutions to exponential differential equations is a key area of analysis, particularly when the growth rate is positive. Even if two solutions start from different initial conditions, the exponential nature of the growth causes their difference to increase without bound over time. This principle is essential for understanding phenomena such as sensitivity to initial conditions and the divergence of trajectories in dynamical systems.

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