Maths with David

    IBDP. AI. HL. DEs. Form f’(x) = g(x)

    The easiest type of DEs to solve are ones where \frac{dy}{dy} = f(x) and f(x) is integrable. In this case we simply integrate both sides of the equation with respect to x.

    Worked Example 1. General Solution

    Find the general solution to (a) \frac{dy}{dx} = e^{2x} , (b) \frac{dy}{dx} = \frac{2x}{x^2+1}

    Worked Example 2. Particular Solution

    Find the particular solution to \frac{dy}{dx} + 1 = cos(x) , given that y(0) = \frac{1}{2}

    Worked Example 3. Economics (Marginal Cost)

    The marginal cost of producing x widgets per week is given by \frac{dC}{dx} = 2.15 - 0.02x + 0.00036x^2 pounds per widget provided 0 \leq x \leq 120 .

    The initial costs before production starts are £185. Find the total cost of producing 100 widgets per day.

    Worked Example 4. Physics (Heat Transfer)

    A metal water pipe has an outer radius of 4cm and inner radius of 2cm. Within the pipe water temperature is maintained at 100ºC. Within the metal, the temperature drops off from inside to outside according to \frac{dT}{dx} = - \frac{10}{x} , where x is the distance from the central axis and 2 \leq x \leq 4 . Find the temperature of the outer surface of the pipe.

    Exercise

    1.) Find the general solution to:

    (a) \frac{dy}{dx} = 4x^3

    (b) \frac{dy}{dx} = x^2 + 6x

    (c) \frac{dy}{dx} = e^{3x} + 4

    (d) \frac{dy}{dx} = cos(x) + sin(2x)

    (e) \frac{dy}{dx} = \frac{1}{x+4}

    (f) \frac{dy}{dx} + \frac{2}{x} = \sqrt{x}

    2.) Find the general solution to:

    (a) \frac{dM}{dt} = \frac{3t^2}{t^3 - 4}

    (b) \frac{dy}{dt} = \frac{t}{ \sqrt{25 - t^2} }

    (c) f^\prime(t) = te^{ -t^2 + 1}+ 2

    3.) Find the particular solution to:

    (a) \frac{dy}{dx} = 3x-2 , given y(0) = 5

    (b) \frac{dy}{dx} = e^{3x} + 1 , given y(0) = 0

    (c) \frac{dy}{dx} = \frac{1}{x} , given y(2) = ln(12)

    4.) Find the particular solution to:

    (a) \frac{dy}{dt} = 2e^{2t} - e^{-t} given that y=2.5 when t = ln(2)

    (b) \frac{dM}{d \alpha } = cos(2 \alpha ) - 3 sin ( \alpha ) given that M = 2 when \alpha = \frac{ \pi}{2}

    5.) A function f(x) has gradient function f’(x) = 2x – 5, and the point (2,-18) lies on the curve. Find the value of f(-2).

    6.) (a) Find the particular solution of \frac{dy}{dx} = e^x - e^{-x} given y(0) = 1

    (b) Sketch the graph of the particular solution

    (c) Find the equation of the tangent to the solution curve at the point where x = ln2.

    7.) A curve has gradient function f^{\prime} (x) = ax + bx^{-2} where an and b are constants. It passes through (-1,-2) and has a turning point at (1,0). Find the function f(x).

    8.) Find the particular solution to:

    (a) f’’(x) = 6x – 4, given f’(1) = 3 and f(2) = 7

    (b) \frac{d^2x}{dx^2} = sin2x given y(0) = 0 and y( \frac{ \pi }{2} ) = 2 \pi

    9.) The marginal cost per day of producing x gadgets is C’(x) = 3.15 + 0.004x pounds per gadget. Find the total daily production cost for 800 gadgets given that the fixed costs before production commences are £450 per day.

    10.) The marginal profit for producing x dinner plates per week is given by P’(x) = 15 – 0.03x pounds per plate. If no plates are made then a loss of £650 each week occurs.

    (a) Find the profit function P(x)

    (b) What is the maximum profit, and when does it occur?

    (c) What production levels enable a profit to be made?

    11.) An insulation tube has inner radius 0.02m and outer radius 0.04m. Fluid flowing through the tube maintains the temperature on the inner wall at 600ºC. Heat is lost through each metre of tube length according to Fourier’s Law: \frac{dT}{dr} = \frac{q}{2 \pi kr} where:

    q = 680Wm-1 is the heat transfer rate per metre of length

    k = 0.2 Wm-1 ºC is the thermal conductivity constant

    r is the radius from the centre of the tube; and

    T is the temperature in ºC.

    Calculate the external temperature of the tube.

    12.) Fluid  flows through a stainless steel pipe with thermal conductivity k = 19 Wm-1 °C. The pipe has inner radius r1 = 0.14m and outer radius r2=0.20 m.

    The inner wall temperature is maintained at 400°C.

    The stainless steel pipe needs to be insulated with urethane foam with k = 0.018 Wm-1°C so that the temperature on the outside of the foam is not more than 50°C.

    The heat loss q = 60 Wm-1 is constant throughout the pipe and the insulation. Use Fourier’s law: \frac{dT}{dr} = \frac{q}{2 \pi kr} to find:

    (a) The temperature of the outer surface of the pipe

    (b) The outer radius r3 of the insulation

    (c) The thickness of insulation needed

    13.) A wooden plank is supported only at its ends O and P, which are 4m apart. The plank sags under its own weight by y m at the distance x m from O.

    The differential equation \frac{d^2x}{dx^2} = 0.01( 2x - \frac{x^2}{2} ) relates the variables x and y , for 0 \leq x \leq 4

    (a) Find the function y(x) which measures the sag from the horizontal at any point along the plank.

    (b) Find the maximum sag from the horizontal. Does the position of the point of maximum sag seem reasonable?

    (c) Find the sag 1m away from P.

    (d) Find the angle the plank makes with the horizontal at the point 1m from P.

    Answers