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 same as in (a) except the determination of the constants.
(5)
(6)
Substitute (5) into (6)

So
The solution is

The blue curve is for part (a), and the green curve is for part (b).

2. Repeat problem 1 for the equation:

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

, thus
(1)
so . Since equation (1) has 2 complex roots, then the general solution of the equation is

For the above given initial conditions
,

So the general solution is

(b) y'' + 2y' + 5y = 0 y(0) = 0; y'(0) = 5
For this pair of initial conditions,

The general solution is

The blue curve is for part (a), and the green curve is for part (b).

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 centimeter sand then striking it hard enough to impact 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

Assume the equilibrium is the origin, and the downward direction is positive x-direction.

The force by the spring is determined by Hook's law, , where x is the displacement of the object in meters.
The force by the shock absorber is
According to Newton's second law,

Where m=1kg, and a is the acceleration of the mass,

So it is a constant coefficient second order differential equation.
Assume
Then the same analysis as in problem (1)

So

Since it has two real roots, then the general solution is
(1)
Then
So the initial displacement

The initial velocity is upward and 5 m/s. so v0= -5m/s, the negtiave sign is due to the downward positive direction.

So
(2)

Now we can solve the above initial conditions of x.
(a) The system is set in motion by lowering the bob 25/3 centimeter, then
x0= 25/3 cm =25/300 = 1/12 m.

(3)
Solving (2) and (3),
Plug c1 and c2 into (1), the displacement function is
(b) with x0=12 cm = 0.12 m
(4)
Solving (2) and (4),
Plug c1 and c2 into (1), the displacement function is

(c) with x0=20 cm =0.2m
(5)
Solving (2) and (5),
Plug c1 and c2 into (1), the displacement function is
(d) with x0=30 cm =0.3m
(6)
Solving (2) and (6),
Plug c1 and c2 into (1), the displacement function is
(e) with x0=45 cm =0.45m
(7)
Solving (2) and (7),
Plug c1 and c2 into (1), the displacement function is

The initial displacements are labeled in the graph. For example, blue line is for 25/3 cm initial displacement, purple is for 45cm displacement.