Differential Equations: Definitions, Classification, And Solutions

(29 MARKS)**

a. Determine the order, degree and linearity of the differential equation

$$
xy'' = \sqrt{y} + y' + y^3
$$

(3 marks)
b. Form the differential equation for (y = a \cos 3x + b \sin 3x) where (a) and (b) are arbitrary constants.
(3 marks)
c. Determine (c_1) and (c_2) so that (y(x) = c_1 \sin 2x + c_2 \cos 2x + 1) will satisfy the conditions

$$
y\left(\frac{\pi}{4}\right) = 0 \quad \text{and} \quad y'\left(\frac{\pi}{8}\right) = \sqrt{2}
$$

(5 marks)
d. Explain what is meant by the Wronskian of the terms (u_1), (u_2), and (u_3), hence use the Wronskian to show that if the solutions of a differential equation are (e^{3x}) and (x^2e^{3x}), then they are linearly independent. Give the general solution of the differential equation.
(8 marks)
e. Solve the differential equation ((\sin x + \cos x)dy + (\cos x - \sin x)dx = 0)
(3 marks)

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$$
(5x^4 + 3x^2y^2 - 2xy^3)\,dx + (2x^3y - 3x^2y^2 - 5y^4)\,dy = 0
$$

is exact. Hence solve it. (5 marks)

c) Find the orthogonal trajectory of the family of (y^2 = 4Cx) (5 marks)

(a) Show that the differential equation

$$
(2x\cos y + 3x^2 y)\,dx + (x^3 - x^2\sin y - y)\,dy = 0
$$

is exact and hence find its general solution. (3 marks)
(b) Determine the general solution of the differential equation

$$
(x^2 + 1)\frac{dy}{dx} + 3xy = 6x.
$$

(5 marks)
(c) The size (P) of the population of a habitation is modelled as a continuous function of the real variable (t), where (t) is in years. The rate of increase of (P) is directly proportional to (P).

(i) Write down a differential equation that is satisfied by (P). (1 mark)
(ii) Given that the initial size of the population is (P_0), show that (P = P_0 e^{kt}), where (k) is a positive constant. (3 marks)
(iii) The initial size of the population at the foundation of the habitation is 15. At 120 years, the population is (P_1 = 75). Determine (P_0) and (k) in (P = P_0 e^{kt}). Give your answer for (k) as an exact logarithm. (3 marks)
(iv) The habitation may be defined to be a metropolis when the population reaches 1,000,000 inhabitants. In what year, after the foundation, will the habitation become a metropolis? (3 marks)