# Differential Equations and Springs

1. Solve the initial-value problems and graph the solutions on the same set of axes.

y'' + 4y' + 2y = 0 y(0) = 5; y'(0) = 0

y'' + 4y' + 2y = 0 y(0) = 0; y'(0) = 5

2. Repeat problem 1 for the equation:

y'' + 2y' + 5y = 0 y(0) = 5; y'(0) = 0

y'' + 2y' + 5y = 0 y(0) = 0; y'(0) = 5

3. An object having a mass of 1 kg. is suspended from a spring with a spring constant (k) of 24 Newtons/meter. A shock absorber which induces a drag od 11v newtons (v is in meters/second) is included in the system.

The system is set in motion by lowering the bob 25/3 centimeters and then striking it hard enough to impaet an upward velocity of 5 meters/sec; solve for and graph the displacement function.

repeat this exercise (all on the same graph) for cases where the bob is lowered:

12, 20, 30, and 45 centimeters.

Please solve all 3 problems

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The differential equations in Problem 1 and 2 are all have constant coefficients. Assume that the general solution is , then , and . Substitute them into the equations to find the solutions. All the following equations are solved using the assumption.

1. Solve the initial value problems and graph the solutions on the same set of axes.

(a) y'' + 4y' + 2y = 0 y(0) = 5; y'(0) = 0

, thus

(1)

so . Since equation (1) has 2 real roots, then the general solution to the differential equation is

, where c1 and c2 are constants, which can be determined by the initial conditions.

(2)

So (3)

Now we need to solve (2) and (3) to find the constants c1 and c2.

From (3)

(4)

Substitute (4) into (2)

Then

So the solution of the differential equation is

(b) y'' + 4y' + 2y = 0 y(0) = 0; y'(0) = 5

All the analysis are the ...

#### Solution Summary

IVPs, Differential Equations and Springs are investigated. The solution is detailed and well presented.