Questions: (a) Solve the initial value problem y''+9y'+18y=0, y(0)=2, y(0)=3 y(t)=(2+9t)e^(-3t) (b) Describe the behavior of the solution as t increases, Your answer should be a number, lim as t approaches infinity of y(t)=0

(a) Solve the initial value problem
y''+9y'+18y=0, y(0)=2, y(0)=3
y(t)=(2+9t)e^(-3t)
(b) Describe the behavior of the solution as t increases, Your answer should be a number,
lim as t approaches infinity of y(t)=0

Transcript text: (a) Solve the initial value problem \[ y^{\prime \prime}+9 y^{\prime}+18 y=0, y(0)=2, y(0)=3 \] \[ y(t)=(2+9 t) e^{-3 t} \] (b) Describe the behavior of the solution as $t$ increases, Your answer should be a number, \[ \lim _{t \rightarrow \infty} y(t)=0 \]

Solution

Solution Steps

Solution Approach

(a) To solve the initial value problem, we need to find the general solution to the differential equation $ y'' + 9y' + 18y = 0 $ and then apply the initial conditions $ y(0) = 2 $ and $ y'(0) = 3 $ to determine the specific solution.

(b) To describe the behavior of the solution as $ t $ increases, we need to find the limit of $ y(t) $ as $ t $ approaches infinity.

Step 1: Solve the Differential Equation

We start with the initial value problem given by the differential equation:

\[ y'' + 9y' + 18y = 0 \]

The general solution to this equation is:

\[ y(t) = (C_1 + C_2 e^{-3t}) e^{-3t} \]

Step 2: Apply Initial Conditions

We apply the initial conditions $ y(0) = 2 $ and $ y'(0) = 3 $ to find the specific solution. After substituting these conditions, we find:

\[ y(t) = (5 - 3e^{-3t}) e^{-3t} \]

Step 3: Analyze the Behavior as $ t $ Increases

To describe the behavior of the solution as $ t $ approaches infinity, we calculate the limit:

\[ \lim_{t \to \infty} y(t) = 0 \]