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Solve the IVP and graph the solution: y ′′ +2y ′ +2y=0y(0)=0,y ′(0)=1 While the solution is not a periodic function, it does cross the axis at regular intervals. So find all the roots of the solution, and compute the distance between successive roots to see that it is a constant.

User Lancy
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The roots of the auxiliary equation are -1 + i and -1 - i.

To solve the given initial value problem (IVP) and graph the solution, we can follow these steps:

Step 1: Write the auxiliary equation
The given differential equation is y'' + 2y' + 2y = 0. To solve it, we can write the auxiliary equation by assuming y = e^(rt), where r is a constant:
r^2 + 2r + 2 = 0

Step 2: Solve the auxiliary equation
To solve the quadratic equation, we can use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 1, b = 2, and c = 2. Substituting these values into the quadratic formula, we get:
r = (-2 ± √(2^2 - 4(1)(2))) / (2(1))
r = (-2 ± √(-4)) / 2
r = (-2 ± 2i) / 2
r = -1 ± i

So, the roots of the auxiliary equation are -1 + i and -1 - i.

Step 3: Write the general solution
Since the roots of the auxiliary equation are complex, the general solution can be written as:
y = C1e^((-1 + i)t) + C2e^((-1 - i)t)

Step 4: Apply initial conditions
We are given the initial conditions y(0) = 0 and y'(0) = 1. Let's use these conditions to find the values of C1 and C2.

Substituting t = 0, y = 0 into the general solution:
0 = C1e^0 + C2e^0
0 = C1 + C2

Next, differentiate the general solution with respect to t:
y' = -C1e^((-1 + i)t) - C2e^((-1 - i)t)

Substituting t = 0, y' = 1 into the derived equation:
1 = -C1e^0 - C2e^0
1 = -C1 - C2

Now, we have a system of equations:
0 = C1 + C2
1 = -C1 - C2

Solving this system, we find C1 = -1/2 and C2 = 1/2.

Step 5: Write the particular solution
Using the values of C1 and C2, we can write the particular solution as:
y = -1/2e^((-1 + i)t) + 1/2e^((-1 - i)t)

Step 6: Simplify the particular solution
To simplify the particular solution, we can use Euler's formula, which states that e^(it) = cos(t) + isin(t).

Using Euler's formula, we can rewrite the particular solution as:
y = -1/2e^(-t)cos(t) + i/2e^(-t)sin(t) + 1/2e^(-t)cos(t) - i/2e^(-t)sin(t)

Simplifying further, we get:
y = e^(-t)cos(t)

Step 7: Graph the solution
To graph the solution, we can plot the function y = e^(-t)cos(t) on a coordinate system.

The distance between successive roots of the solution can be computed by finding the values of t where y = 0. In this case, the roots occur at regular intervals since the function is not periodic. By calculating the difference between consecutive root values, we can verify if it is constant.

Solve the IVP and graph the solution: y ′′ +2y ′ +2y=0y(0)=0,y ′(0)=1 While the solution-example-1
User IWizard
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