Q1. Identify Ordinary Differential Equations (ODEs) and write the ODEs in the form of either F(t,x,x^·) = 0 or F(t,x,x^·,x^··) = 0

a. τx^·(t) + x(t) = k
b. 2x² + 4x + 1 = 0
c. dx/dt = -2x + 3xt
d. d²x/dt² + λ²x = 0
e. ∂²u(x₁, x₂, x₃)/∂x₁² + ∂²u(x₁, x₂, x₃)/∂x₂² + ∂²u(x₁, x₂, x₃)/∂x₃² = 0
f. mx^·· + cx^· + kx = f(t)

Q2. Give a summary of the main steps of doing mathematical modelling for one engineering system that you know.

Q3. Classify the following DEs as linear homogeneous, linear nonhomogeneous or nonlinear differential equations. Also state their order and name the dependent and independent variables.

a. d²s/dt² + (sin(t)) ds/dt + (t + cos(t))s = e^t
b. d²x/dt² + dx/dt - 4x = cos(t) + e^t
c. d³p/dy³ + 4d²p/dy² - 2p(dp/dy)² = sin(y)
d. (dx/dt)² + x = 0

Q4. Verify that x = ce^{-t^3/2}, where c is an arbitrary constant, is a solution of the following ODE. Also find the particular solution for the initial-value problem.

2t.dx/dt + 3t³ x = 0, x(0) = -1

Q5. Verify that the following first-order ODE is of separable form. Then solve it to obtain a general solution.
t sinxdt + (t² + 1) cos xdx = 0

Q6. Verify that the following first-order ODE belongs to the form of dx/dt = f(x/t). If so, solve it to obtain the general solution.

ktx dx/dt = x² - t².

where k> 0 and k ≠ 1 is a constant.

After that, considering the case when k is the summation of your student ID (e.g. if your student ID is 34567 the value of k will be 3+4+5+6+7=25). Write your general solution for the correspond¬ing value of k.

Q7 Test that the following first-order ODE is of exact form (this test is also called the exactness test). If so, find the general solution.

(2x + 3t)dx/dt + 3x + 2f = 0.

Q8 Verify that the following first-order ODE is a linear differential equation. If so, solve for its initial-value problem.

dx/dt - (3 - 2x)/t = 4t + e^t, x(1) = 0

Q9 Using the Eulers method with a step size of h = 0.1, find the value of X(0.3) for the initial-value problem

dx/dt= 3x - 2t, x(0) = 1

Q10 Find the general solution of the following differential equation:

d²x/dt² - 4dx/dt + 7x = e-^3t

Q11 Find the general solution of the following differential equation:

d²x/dt² + 2dx/dt -3x = 5e^-t + sin 3t 

Some guidance:

- All of the questions in this Assignment 1 are related to Topics 1 to 5.

- Except Question 6, ten out of eleven questions are of similar types as the ones you have dealt with in Seminars of Topics 1 to 5. If you practice sufficiently solving the seminar questions, you can solve these seminar-type questions.

- Question 6 is a slightly challenging question. It is designed for whom is aiming to get Distinction and High Distinction in Assignment 1. So, if you do not know how to solve Question 6 at first sight, leave it to the last. Solve the other questions first and then come back to tackle the Question 6 afterwards.

- Even Question 6 can be considered similar to the one you have learnt in Topic 3. It is of the form dx/dt = f (x/t).

- All of the eleven questions can be solved using the knowledge you have learnt from Topics 1 to 5.