×
Get Full Access to Elementary Differential Equations And Boundary Value Problems - 10 Edition - Chapter 11.1 - Problem 23
Get Full Access to Elementary Differential Equations And Boundary Value Problems - 10 Edition - Chapter 11.1 - Problem 23

×

# Consider the problemy+ y = 0, y(0) = 0, y(L) = 0.Show that ISBN: 9780470458310 168

## Solution for problem 23 Chapter 11.1

Elementary Differential Equations and Boundary Value Problems | 10th Edition

• Textbook Solutions
• 2901 Step-by-step solutions solved by professors and subject experts
• Get 24/7 help from StudySoup virtual teaching assistants Elementary Differential Equations and Boundary Value Problems | 10th Edition

4 5 1 424 Reviews
29
4
Problem 23

Consider the problemy+ y = 0, y(0) = 0, y(L) = 0.Show that if m and n are eigenfunctions corresponding to the eigenvalues m and n,respectively, with m = n, thenL0m(x)n(x) dx = 0.Hint: Note thatm + mm = 0, n + nn = 0.Multiply the first of these equations by n, the second by m, and integrate from 0 to L,using integration by parts. Finally, subtract one equation from the other.

Step-by-Step Solution:
Step 1 of 3

PHYS Notes Week 7 Feb 22­26 Electric Charge ­ Particles have either a positive or negative charge ­ Combining these particles into atoms/molecules result in three possibilities ­ Negatively charged: object contains more negative particles than positive particles ­ Positively charged: object contains more positive particles than negative particles ­ Electrically neutral: object contains equal amounts of positive and negative particles ­ Nature prefers neutral charges ­ The terms "positive" and "negative" don't mean anything; they just refer to the fact that the charges are opposite ­ Electrostatic/electric force: the force that charged particles exert on each other ­ Objects with the same electrical charge repel each other, while objects with opposite electrical charges attract each other ­ Strong electrical charges can induce an opposite charge in a neutrally charged system ­ Grounding it can neutralize a system’s charge ­ Grounding: touching an object to the ground (the earth is so big that it can absorb any extra charge without problem) ­ Unit of electric charge is a coulomb (C) ­ Derived from base unit of ampere, which is a measure of current ­ Current: rate at which charge moves past a given point in a given amount of time ­ Charge is quantized (comes in basic units based on electrons that cannot be divided) ­ Basic unit of charge: electron ­ Historically led to the development of quantum mechanics ­ Charge is conserved (cannot be created or destroyed, only moved around) ­ Charge moving through materials ­ Conductors vs. insulators ­ Conductors allow electrons to move freely (ex. metal) ­ Everything can be a conductor with enough electricity ­ Insulators don't allow electrons to move as freely ­ Semi­conductors are somewhere in between ­ Superconductors allow charge to move without hindrance ­ Coulomb’s law ­ Force exerted by charged particles on each other depends on the size of the charge of the particles as well as their distance from one another ­ Two positive or two negative charges ­­> particles push away from each other; one negative and one positive charge ­­> particles attract each other * Increasing force means opposite charges (attracting particles) ­ Electric fields: electrostatic forces existing around a charged particle ­ Determine what a field looks like by placing a test charge near it and measuring the force applied to the test charge ­ We draw electric field using field lines ­ Field lines closer together shows stronger force ­ Field lines extend away from positive charges and toward negative charges ­ If we create a field, we can direct a particle through it (old TVs) ­ Charged particles have potential energy ­ Electric potential (voltage): potential energy per electrical charge ­ Electric current: flow of electrons in motion (negative to positive) ­ Produced by voltage ­ Inserting battery into loop of conductive material creates a flow ­ Conduction ­ Some materials conduct electricity better due to resistance of the material ­ Resistance inhibits flow ­ Ohm: unit of resistance ­ Ohm's law: as potential increases, current increases, and when resistance increases, current decreases ­ Series Circuits ­ Battery/power supply: creates a difference in potential energy ­ Path from one end of battery to another (wire or other conductive material) ­­> electrical current ­ Electrons pushed through a resistor, which slows the current down/steals kinetic energy from the electrons to power a machine ­ Power multiple machines by adding multiple resistors to circuit around the circuit ­ Each resistor increases the overall resistance of circuit ­ Parallel circuits ­ Put resistors into a circuit next to each other, creating multiple paths for the electron to move through ­ If one of the paths slows down (because electrons have to slow to enter the resistor), the backed­up electrons move through the next parallel resistor ­ Adding resistors decreases the resistance of the circuit, increasing current flow (like adding lanes to a highway) ­ Too fast of a current is an issue because wires can only hold so much electricity ­ Direct current (DC): current that flows in only one direction ­ Usually used in electronics/devices ­ Alternating current (AC): current alternates direction (60 times/s (60 Hz)), which changes, faster than what we can see (20 Hz) ­ Easier to generate and travel over long distances ­ Argument between Tesla (AC) vs. Edison (DC) because AC is dangerous ­ Edison created the electric chair, which used AC ­ Transformers are used to change between AC and DC

Step 2 of 3

Step 3 of 3

##### ISBN: 9780470458310

Unlock Textbook Solution