Review

Definition of the Derivative

f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}

The derivative at a point is the slope of the tangent line to the graph of the function at that point.

Derivative Formulas

  1. Constant Rule: f(x) = c \quad f'(x) = 0

  2. Constant Multiplier Rule: f(x) = c \cdot g(x) \quad f'(x) = c \cdot g'(x)

  3. Power Rule: f(x) = x^n \quad f'(x) = nx^{n-1}

  4. Sum Rule: f(x) = g(x) + h(x) \quad f'(x) = g'(x) + h'(x)

  5. Product Rule: f(x) = g(x) \cdot h(x) \quad f'(x) = g(x) \cdot h'(x) + g'(x) \cdot h(x)

  6. Quotient Rule: f(x) = \frac{g(x)}{h(x)} \quad f'(x) = \frac{h(x) g'(x) - h'(x) g(x)}{[h(x)]^2}

  7. Chain Rule: f(x) = g \circ h (x) \quad f'(x) = g' \circ h (x) h'(x)

Differential Equations

We will be dealing with linear differential equations with constant coefficients only.

Homogeneous Solution

To find the homogeneous solution to a differential equation, find the roots of the characteristic polynomial. Recall that the characteristic polynomial is formed by sending y^{(n)} to r^n where y^{(n)} denotes the nth derivative of y and y^{(0)} = y. Then use the table below to form the homogeneous solution, y_h, for your differential equation:

Roots

Solutions

one root: r

y_h = Ae^{rt}

two distinct roots: r1, r2

y_h = Ae^{r_1 t} + Be^{r_2 t}

one repeated root: r

y_h = Ae^{r t} + Bte^{r t}

two complex roots: a \pm bi

y_h = Ae^{a t} \cos(bt) + Be^{a t} \sin (b t)

Particular Solution

To find a particular solution to a differential equation, make a guess with undetermined coefficients based on the right-hand side of your equation. Then plug the guess back into your differential equation and determine the coefficients. The following table for guesses is helpful:

Right-hand Side

Guess

a_n t^n + ... + a_1 t + a_0 (a_n \neq 0)

b_n t^n + ... + b_1 t + b_0

Ae^{at}

Be^{at}

C \sin(at) + D \cos(at)

A \sin(at) + B \cos(at)

Remember that if your guess "doesn't work", i.e. it is a homogeneous solution (looks like y_h), then you multiply your guess by t.

General Solution

The general solution is the sum of the homogeneous and particular solutions: y = y_h + y_p

Initial Values

If initial values are given, then use them to determine the coefficients in your general solution.

Practice Exercises

Using the derivative formulas above, compute each of the following derivatives in questions 1-10. You may replace the letters serving as constants with your favorite numbers (a,b,c,d,e, and f) first if you wish. In particular, note that making some constants equal to one will greatly simplify the problem.

  1. y(x) = ax^b

  2. y(x) = a(b + cx^d)^e

  3. y(x) = \frac{ax^b + c}{dx^e + f}

  4. y(x) = a

Past Quizzes

Quiz

Solutions

LaTeX source

quiz1.pdf

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quiz2.pdf

quiz2-solutions.pdf

quiz2.tex

quiz3.pdf

quiz3-solutions.pdf

quiz3.tex

quiz4.pdf

quiz4-solutions.pdf

quiz4.tex

quiz5.pdf

quiz5-solutions.pdf

quiz5.tex

quiz6.pdf

quiz6-solutions.pdf

quiz6.tex

Questions

How do I ask a question?

NUWiki: Math1251 (last edited 2011-08-07 18:22:05 by JasonRibeiro)