Which of the methods of ﬁnding analytic solutions of diﬀerential equations described in Unit 2 could you use to solve this equation? Give reasons for your answer.

(b) Find the general solution of the diﬀerential equation, expressing y explicitly as a function of x. Hence ﬁnd the particular solution of the diﬀerential equation that satisﬁes the initial condition y(1) = 1.

Question 3.

You must solve this problem by hand, and the solution that you submit to your tutor should contain all your working.

(a) Consider the diﬀerential equation tan(2t)dx/dt = x^2 (0 < t 0).

Which of the methods of ﬁnding analytic solutions of diﬀerential equations described in Unit 2 could you use to solve this equation? Give reasons for your answer.

(b) Find the general solution of the diﬀerential equation, expressing x explicitly as a function of t. Hence ﬁnd the particular solution of the diﬀerential equation that satisﬁes the initial condition x(π8)=1/2

Question 4

This question is concerned with the use of Euler’s method to ﬁnd a numerical solution to the initial-value problem dy/dx = x^2 − sin(y), y(0) = 0.

In part(a) you may use a computer or calculator only to perform numerical calculations. In part (b), on the other hand, you are expected to use one of the Mathcad worksheets. You may ﬁnd it helpful to use the same worksheet in part (c).

(a) Explain why this diﬀerential equation cannot be solved analytically by the methods described in Unit 2.

(b) Use Euler’s method with a step size of 0.1 to ﬁnd an approximation to the value of y(0.3), where y(x) is the solution to the given initial-value problem. Carry out your calculations using at least six decimal places. Show all your working, and quote your ﬁnal answer to four decimal places.

Question 5

In parts (a)–(c) you must solve the problem by hand, and you must show your working in your solution. In part (d) you are expected to use a computer.

(a) Determine the general solutions of the following linear second-order homogeneous diﬀerential equations.

(i)d^2y/dx^2+ 5dy/dx + 4y = 0

(ii) d^2y/dx^2+ 6dy/dx + 9y = 0

(iii) d^2y/dx^2+ 6dy/dx + 10y = 0

(b) Find a particular integral for the inhomogeneous diﬀerential equation d^2y/dx^2+ 6dy/dx + 10y = −6e−2x + 10x.

Hence write down the general solution of this equation.

(c) Find the particular solution to the initial-value problem

in this question you should quote all numerical answers correct to three signiﬁcant ﬁgures. However, intermediate values should be given to at least ﬁve signiﬁcant ﬁgures. Any symbols that you introduce must be deﬁned.

A rectangular awning OABC is attached to a vertical wall between the points O and A, which are at the same horizontal height. The awning is 4 m wide and 2 m deep, and is inclined downwards at an angle of 70◦to the vertical, with the outer edge BC below the line of attachment to the wall. The point P is on OA at three-quarters of the distance between O and A. 3 m vertically below point P is the point D, which is the support forapole that is attached to the point B and extends upwards to the point E, which is in the same horizontal plane as OA.

The perspective diagram below attempts to illustrate the arrangement.

(a) Taking the origin at O and axes as shown in the ﬁgure (i is horizontal and within the plane of the wall, j is vertically upwards, and k is horizontal and perpendicular to the wall), ﬁnd the position vectors of the points A and C. Also ﬁnd −−→OB.

(b) Find the position vector of the point D. Hence determine the vector equation of the line DB, and hence ﬁnd the position vector of the point E.

(c) Find the total length of the pole DE.

(d) Find the dot product of the vectors −−→BA and −−→BE, and hence determine the angle ABE. Give your answer in degrees.

(e) The triangle ABE is to be ﬁlled in by an extra awning. Find the cross product of the vectors −−→BA and −−→BE, and use this result to determine the area of the triangle ABE.

Question 7

Consider the vector v shown in the following diagram. Find the i- and j-components of v in terms of the magnitude of v, α and θ, simplifying your answers as far as possible, and explain your method of approach.

PLEASE THIS IS MATHEMATICS QUESTIONS SO I NEED MATHEMATICS EXPERT NOT LAW .

THANKS

I couldn't copy the diagrams so I would like the expert to tell me any other way i can send.