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A vibration of a plate is described as even tho + 13y = 26* +cos x...
Question 1 LO1, PO1 A deflection of a beam is described as dy dy 40d 13y=re...
Question 1 LO1, PO1 A deflection of a beam is described as dy dy 40d 13y=re dr? dx 1) Solve the general solution. [32 marks] i) Calculate the relationship between y and x using initial conditions | x = 0, y = 0, "v/dt = 0] [8 marks] Enter your answer ON 99+
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy....
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0
4. (a Let (sin( x cos( ) dr...
The vibration of a semi-infinite string is described by the following initial boundary value problem.
(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.$$ \begin{array}{l} u_{t t}=c^{2} u_{x x}, \quad 0< x < \infty, t>0 \\ u(x, 0)=A e^{-\alpha x} \quad \text { and } \quad u_{t}(x, 0)=0, \quad 0< x < \infty \\ u(0, t)=A \cos \omega t, \quad t>0 \\ \lim _{x \rightarrow \infty} u(x, t)=0, \quad \lim _{x...
1. Find the particular solution of the differential equation dydx+ycos(x)=2cos(x)dydx+ycos(x)=2cos(x) satisfying the initial condition y(0)=4y(0)=4. 2. Solve...
1. Find the particular solution of the differential
equation
dydx+ycos(x)=2cos(x)dydx+ycos(x)=2cos(x)
satisfying the initial condition y(0)=4y(0)=4.
2. Solve the following initial value problem:
8dydt+y=32t8dydt+y=32t
with y(0)=6.y(0)=6.
(1 point) Find the particular solution of the differential equation dy + y cos(x) = 2 cos(z) satisfying the initial condition y(0) = 4. Answer: y= 2+2e^(-sin(x)) Your answer should be a function of x. (1 point) Solve the following initial value problem: dy ty 8 at +y= 32t with y(0) = 6. (Find y as...
The position x of a mass m attached to a spring obeys the differential equation i...
The position x of a mass m attached to a spring obeys the differential equation i + yi + w?x = 0 where y 2w. a) (2 marks) Write down expressions for the forces on the mass due to (i) the spring, and (ii) damping. (3 marks) Using a trial solution x = Ae"', show that a = --y/2 ± (y2/4 - «2)1/2 b) c) (4 marks) Show, by finding wd, that the solution is a damped oscillation of the...
(c) (i) Find the general solution of the following partial differential equation y, = 2y sin...
(c) (i) Find the general solution of the following partial differential equation y, = 2y sin x + e-x Whatische solution when the initial conditions are v(0,y)--y, and (ii) y(x, 0) = cos x ? (10 Marks)
Solve i. and ii. Given the ordinary differential equation: cos(x)y' = sin(x)y + 1 Find the...
Solve i. and ii. Given the ordinary differential equation: cos(x)y' = sin(x)y + 1 Find the general solution of the given differential equation. ii. Solve the ordinary differential equation: ay' + by = a cos(wx) + Bsen(wx) Where: a, b, a,ß and w are nonzero real constants.
3.1 The relationship between the input x(t) and output y(t) of described by the indicated differential...
3.1 The relationship between the input x(t) and output y(t) of described by the indicated differential equation given below: a causal system is dx(t) dse)+540+6y(t) = x(t) +T Assuming that the initial conditions are zero and using the Laplace transform determine [5 Marks] 15 Marks the following: a- Transfer function H(s) of the system. b- Impulse response h(t) of the system. Y (s) X(s)
Torsional vibration of a shaft is governed by the wave equation, = 16 where (x,t) is the angular displacement (angle of...
Torsional vibration of a shaft is governed by the wave equation, = 16 where (x,t) is the angular displacement (angle of twist) along the shaft, ar is the distance from the end of the shaft and t is time. For a shaft of length 2T that supported by frictionless b end, the boundary conditions are 0r(0,t) = 0x(2T, t) = 0, t> 0. Suppose that the initial angular displacement and angular velocity are (x,0) = 6 cos(x), Ot(x,0) =3+2 cos(42),...
An LTI system is described by the following differential equation. Find the output when x(t)- u(t)...
An LTI system is described by the following differential equation. Find the output when x(t)- u(t) and has the following initial conditions: y(0)= 1, (0) = 2 , and x(0)--I dy x dx +at + 4 y(t) = dt + x(t) Solution
Question 1 LO1, PO1 A deflection of a beam is described as dy dy 40d 13y=re dr? dx 1) Solve the general solution. [32 marks] i) Calculate the relationship between y and x using initial conditions | x = 0, y = 0, "v/dt = 0] [8 marks] Enter your answer ON 99+
4. (a Let (sin( x cos( ) dr + (x cos(x + y) - 2) dy. dz= Show that dz is an exact differential and determine the corresponding function f(x,y) Hence solve the differential equation = z sin( Cos( y) 2 x cos( y) dy 10] (b) Find the solution of the differential equation d2y dy 2 y e dx dæ2 initial conditions th that satisfi 1 (0) [15] and y(0) 0
4. (a Let (sin( x cos( ) dr...
1. Find the particular solution of the differential
equation
dydx+ycos(x)=2cos(x)dydx+ycos(x)=2cos(x)
satisfying the initial condition y(0)=4y(0)=4.
2. Solve the following initial value problem:
8dydt+y=32t8dydt+y=32t
with y(0)=6.y(0)=6.
(1 point) Find the particular solution of the differential equation dy + y cos(x) = 2 cos(z) satisfying the initial condition y(0) = 4. Answer: y= 2+2e^(-sin(x)) Your answer should be a function of x. (1 point) Solve the following initial value problem: dy ty 8 at +y= 32t with y(0) = 6. (Find y as...
The position x of a mass m attached to a spring obeys the differential equation i + yi + w?x = 0 where y 2w. a) (2 marks) Write down expressions for the forces on the mass due to (i) the spring, and (ii) damping. (3 marks) Using a trial solution x = Ae"', show that a = --y/2 ± (y2/4 - «2)1/2 b) c) (4 marks) Show, by finding wd, that the solution is a damped oscillation of the...
(c) (i) Find the general solution of the following partial differential equation y, = 2y sin x + e-x Whatische solution when the initial conditions are v(0,y)--y, and (ii) y(x, 0) = cos x ? (10 Marks)
Solve i. and ii. Given the ordinary differential equation: cos(x)y' = sin(x)y + 1 Find the general solution of the given differential equation. ii. Solve the ordinary differential equation: ay' + by = a cos(wx) + Bsen(wx) Where: a, b, a,ß and w are nonzero real constants.
3.1 The relationship between the input x(t) and output y(t) of described by the indicated differential equation given below: a causal system is dx(t) dse)+540+6y(t) = x(t) +T Assuming that the initial conditions are zero and using the Laplace transform determine [5 Marks] 15 Marks the following: a- Transfer function H(s) of the system. b- Impulse response h(t) of the system. Y (s) X(s)
Torsional vibration of a shaft is governed by the wave equation, = 16 where (x,t) is the angular displacement (angle of twist) along the shaft, ar is the distance from the end of the shaft and t is time. For a shaft of length 2T that supported by frictionless b end, the boundary conditions are 0r(0,t) = 0x(2T, t) = 0, t> 0. Suppose that the initial angular displacement and angular velocity are (x,0) = 6 cos(x), Ot(x,0) =3+2 cos(42),...
An LTI system is described by the following differential equation. Find the output when x(t)- u(t) and has the following initial conditions: y(0)= 1, (0) = 2 , and x(0)--I dy x dx +at + 4 y(t) = dt + x(t) Solution
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