Tutorials

Tutorial 1: Review of Algebra and Calculus

Outcomes: At the end of this tutorial you should be able to

- Understand the rules of the natural exponent and logarithmic functions.
- Solve equations containing these functions.
- Differentiate and integrate functions using the associated rules.
- Differentiate and integrate functions using the associated methods

Logarithmic Equations

Solve each of the following logarithmic equations for \(x\):

  1. \(\ln{(5x+4)}=0\)

  2. \(\ln{x}+\ln{0.2}=\ln{e}\)

  3. \(\ln{4x^2}-\ln{16}=\ln{4}-\ln{2x}\)

Revision of Differentiation and Integration Techniques

Differentiate the following functions:

  1. \(x(7x+8)^{2}\)

  2. \(\dfrac{5x^{2}}{(2x^{3}+4)^{4}}\)

Integrate the following functions:

  1. \((2x+1)\ln(x+1)\)

  2. \(x e^{x^{2}}\)

Multiple Choice Questions

  1. If \(f(x)=(x^2-3x)^6(4-x)^5\), then \(f'(x)\) is equal to:

\(\text{(A) } (x^2-3x)^5(4-x)^4(5x^2-21x+24)\)

\(\text{(B) } -(x^2-3x)^5(4-x)^4(7x^2-51x+72)\)

\(\text{(C) } -(x^2-3x)^5(4-x)^4(17x^2-81x+72)\)

\(\text{(D) } 6(x^2-3x)^5(2x-3)(-5)(4-x)^4\)

\(\text{(E) none of these}\)

2) If \(y=e^x \ln{x}\), then \(\dfrac{dy}{dx}\) is equal to:

\(\text{(A) } \dfrac{e^x(1+x\ln{x})}{x}\)

\(\text{(B) } e^x(x+\ln{x})\)

\(\text{(C) } xe^x\)

\(\text{(D) } \dfrac{x}{e^x}\)

\(\text{(E) none of these}\)

3) If \(\dfrac{500}{12+5e^{-0.5x}}\), then \(\dfrac{dy}{dx}\) is equal to:

\(\text{(A) } 500(-1)(12+5e^{-0.5x})^{-2}\)

\(\text{(B) } 500(-1)(12+5e^{-0.5x})^{-2}(12+5e^{-0.5x})\)

\(\text{(C) } 500(-1)(12+5e^{-0.5x})^{-2}(-2.5e^{-0.5x})\)

\(\text{(D) } 500(-1)(5(0.5)e^{-0.5x})^{-2}\)

\(\text{(E) none of these}\)

4) \({\displaystyle \int \dfrac{x^2+4x-\sqrt{x}}{x^2} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } x+\ln{4x}-\frac{2}{3}x^{\frac{3}{2}}+c\)

\(\text{(B) } x+\ln{4x}+\dfrac{2}{\sqrt{x}}+c\)

\(\text{(C) } x+4\ln{x}+\dfrac{1}{2\sqrt{x}}+c\)

\(\text{(D) } x+4\ln{x}+\dfrac{2}{\sqrt{x}}+c\)

\(\text{(E) none of these}\)

5) \({\displaystyle \int \left(\dfrac{3x}{2}-\dfrac{9}{4}\right) e^{x(x-3)} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } \dfrac{3}{4}e^{x^2-3x}+c\)

\(\text{(B) } e^{x^2-3x}+c\)

\(\text{(C) } \dfrac{9}{4}e^{x^2-3x}+c\)

\(\text{(D) } \dfrac{3}{2}e^{x^2-3x}+c\)

\(\text{(E) } \dfrac{4}{3}e^{x^2-3x}+c\)

6) \({\displaystyle \int (3x+1)^{2} e^{-2x} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } -0.5(3x+1) e^{-2x}+1.5(3x+1)^{2} e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(B) } -0.5(3x+1)^{2} e^{-2x}+1.5(3x+1) e^{-2x}+2.25 e^{-2x} + c\)

\(\text{(C) } -0.5(3x+1)^{2} e^{-2x}-1.5(3x+1) e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(D) }-0.5(3x+1)^{2} e^{-2x}+1.5(3x+1) e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(E) none of these}\)

Tutorial 2: ODE’s and Classifcation

Outcomes: At the end of this tutorial you should be able to

- Understand the meaning of an ordinary differential equation.
- Know why classification of an ODE is useful.
- Understand the ways that an ODE can be classified.
- Classify a given differential equation.

Conceceputal Questions

  1. Explain the ways in which one can classify a differential equation.

  2. What does the degree of a differential equation inform us about its linearity?

Classifcation of ODE’s

Classify the following differential equations by checking first if they are ODE’s, and then in terms of their order, linearity, homogeneity, and coefficients. Give reasons for your answers

  1. \(t \dfrac{d^3 x}{dt^3}-2\left(\dfrac{d x}{dt}\right) ^4+x=0\)

  2. \(y y'+2y=1+x^2\), where \(y=y(x)\)

  3. \(y''+9y=\sin y\), where \(y=y(x)\)

  4. \(\dfrac{d^2 R}{dt^2}=\dfrac{\kappa}{R^2}\), where \(\kappa\) (read: kappa) is a constant

  5. \(x^5\dfrac{d^4 y}{dx^4}-x^3 \dfrac{d^3 y}{dx^3}+6y=0\)

  6. \(\dfrac{d^2 x}{dt^2}=\sqrt{1+\left( \dfrac{d x}{dt}\right) ^2}\)

Tutorial 3: Solutions of ODE’s and the Method of Direct Integration

Outcomes: At the end of this tutorial you should be able to

- Understand how to verify a solution of an ODE.
- Sketch a family of solution curves.
- Solve a given ODE using direct integration.

Solutions of ODE’s

  1. In the questions below, verify that the indicated function is an explicit solution of the given differential equation, where \(y=y(x)\).
      1. \(2y'+y=0; \quad y= e^{-x/2}\)
      1. \(y''+y=\tan x; \quad y=-(\cos x)\ln(\sec x+\tan x)\)

  2. In the question below, verify that the indicated family of functions is a solution of the given differential equation.

        \(x^3\dfrac{\textrm{d}^3 y}{\textrm{d}x^3}+2x^2 \dfrac{\textrm{d}^2 y}{\textrm{d}x^2}-x\dfrac{\textrm{d} y}{\textrm{d}x}+y=12x^2; \quad y=c_1x^{-1} +c_2x+c_3x \ln x+4x^2\),

        where \(c_1\), \(c_2\), \(c_3\) are arbitrary constants.

3. Draw a rough sketch or use a graphing calculator (like Desmos) to find the family of solution curves for the differential equation \(y'=\cos x; \quad y=\sin x + c\), where \(y=y(x)\).

Solutions to Differential Equations

Identify the appropriate method of solution, and hence find the solutions of the following differential equations.

  1. \(t \dfrac{d^3 x}{dt^3} = t^4 +t^{-2}\)
  2. \((1+x)dy-x dx=0\), where \(y=y(x)\)
  3. \(e^{-x} \dfrac{d^2 y}{dx^2} = 3\)

Tutorial 4: Method of Separation of Variables and Initial Value Problems

Outcomes: At the end of this tutorial you should be able to

- Solve a given ODE using separation of variables.
- Solve an ODE, and hence find a particular solution given initial conditions.

Solutions to Differential Equations

Identify the appropriate method of solution, and hence find the solutions of the following differential equations.

  1. \(\dfrac{d y}{d x}=\dfrac{e^{x}}{2y}\)
  2. \(\dot{s}+2s=s t^2\), where \(s=s(t)\)
  3. \((e^{2s}-s) \cos r \dfrac{d s}{d r}=e^s \sin 2r\)
  4. \((1+x)dy-y dx=0\)
  5. \(\dfrac{d y}{dx}=\dfrac{x}{y} - \dfrac{x}{1+y}\)

Inital Value Problems

In the problems below, find an explicit solution for the given initial-value problem.

  1. \(\dfrac{\textrm{d} x}{\textrm{d}t}=4(x^2+1); \quad x(\pi/4)=1\)
  2. \(\dfrac{\textrm{d} y}{\textrm{d}x}=\dfrac{y-1}{x-1}; \quad y(2)=2\)
  3. \(\dfrac{\textrm{d}^3 y}{\textrm{d}x^3}=\sin 2x+e^{-2x}+x^2+x+7, \quad \text{subject to} \quad y(0)=0,y'(0)=1,y''(0)=0\)

Tutorial 5: Method of Undetermined Coefficients - Homogeneous Linear DE’s with Constant Coefficients

Outcomes: At the end of this tutorial you should be able to

- Understand when the method of undetermined coefficients is applicable.
- Find the corresponding solution to the homogenous equation, the complementary function.
- Find a particular solution, given initial conditions.

Conceptual Questions

  1. Why does it make intuitive sense to impose a trial solution of the form \(y_{c}(x) = Ae^{\lambda x}\) onto a second-order, linear, homogeneous differential equation with constant coefficients \(a y'' + b y' + c y = 0\), where \(y=y(x)\), and \(a\), \(b\), and \(c\) are independent of \(y\) and \(x\)?

Homogeneous Linear DE’s with Constant Coefficients

Find the general solution to the following differential equations, and hence verify your solution. When an initial condition is given, find the particular solution satisfying the condition/s.

  1. \(y'' -10 y'+25y = 0\), where \(y=y(x)\)
  2. \(8y'' + 4y' + y = 0, \quad y(0) = 1, \quad y'(0) = 0\), where \(y=y(x)\)
  3. \(\ddot{x} + 2 \dot{x} -3 x = 0, \quad x(0) = 0, \quad \dot{x}(0) = 1\), where \(x=x(t)\)
  4. \(\dfrac{d^3 y}{dx^3}-8y=0\)
  5. \(\dfrac{d^2 S}{dt^2}-2\dfrac{dS}{dt}+5S=0\)