Use eulers method or eulers improved method find the

Question

Question 1. A block slides down a rough incline as shown in the diagram below. The frictional force is calculated to be 150 N and the force due to air resistance is 0.5v, where v is the velocity of the block. The weight acting down the slope (parallel to the slope surface) is 250N.

By applying Newton's 2nd Law of motion i.e ∑F=m dv/dt' show that the motion of the block can be modelled by the differential equation

dv/dt=2-1/100 v

Where v,t ≥0

∑F=ma=m (d² x)/(dt² )=mdv/dt

m=50kg

F=m a=dvm/dt

a. Determine the general solution of the differential equation

b. Determine the particular solution if it is known that at t = 0, the object is initially at rest

Question 2. The rate of cooling of a body is given by

dθ/dt=kθ

Where k is a constant. If θ = 55°C when t = 0 and θ = 25°C when t = 5 minutes, determine the time for the temperature to fall to 15°C.

Question 3. A first order differential equation is defined as follows:

dy/dx=11+y

a. Use Euler's Method or Euler's improved method, find the particular solution of the differential equation

b. By using the separation of variables method, find the particular solution of differential equation

c. By using the integrating factor method, verify the particular solution you obtained in part (b)

d. By plotting the results of the numerical method along with the results from one of the analytical methods on the same graph, compare both sets of results.

Question 4. The deflection of a cantilever of length L at a distance x from the fixed end when supporting a uniformly distributed load of W per unit length is given by:

EI (d² y)/(dx² )=1/2 w(L-x)² where x,y ≥0

With initial conditiony(0) = 0 and y'(0)= 0

Find the particular solution of the differential equation

Question 5. An object moves in a straight line so that its distance ,s, metres from the origin after time t seconds is given by;

(s)^.. -8s ?+17s=0

Solve the differential equation given that the initial conditions are s(0)=-4 and s ?(0)=-1

Question 6. Obtain the general solution of the differential equation;

(d² y)/(dx² )+7 dy/dx+12y=e^-2x

Find the particular solution if the initial conditions are y(0)=0 and y'(0)=0

Question 7. A capacitor C is charged by applying a steady voltage E through a resistanceR. The potential difference between the plates,V is given by the differential equation.

CR dv/dt+V=E

a. Solve the equation for V given that V(0) = 0

b. Evaluate the value of the voltage V when E = 55V,C = 5 µF,R = 500 k?& t=4s
E=55V
C=5µF
R=500k?
t=4s

Question 8. A damped oscillation with no external forces can be modelled by the equation:

(d² x)/(dt² )-10 dx/dt+34=0

Where x (in mm) is the amplitude of the oscillation at time tseconds.

The initial amplitude of the oscillation is 2mm (i.e when t = 0) and the initial velocity is0cm/s.

By using the information provided above, determine the particular solution of the differential equation.