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Solution: In 10–13, use the vectorized Euler method with h =

Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider ISBN: 9780321747730 43

Solution for problem 11E Chapter 5.3

Fundamentals of Differential Equations | 8th Edition

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Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Fundamentals of Differential Equations | 8th Edition

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Problem 11E

In 10–13, use the vectorized Euler method with h = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

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Chapter 4: Federalism and The U.S. Constitution 1. Legislature – the body of government that makes laws, because America is a democracy this is the lawmaking power chosen. 2. Bicameral Legislature – this is what the US congress is because it has two separate chambers and the people choose the legislators within the chambers for terms of 2 years [house] or 6 years [senate]. 3. Republic – a government in which decisions are made through representation of the people  Article I Section 8: Spells out the specific powers of congress  Congress can do anything “necessary and proper” to carry out its duties  The House of Representatives: representation based on population, the founders wanted this house to be easily accessible and influenced by the people. [Must be 25 years old and a citizen for 7 years – 2 year term]  The Senate: very elite group, much older, wiser, and more stable than those in the house. [Must be 30 years old and a citizen for 9 years – 6 year term] 4. Unicameral Legislature – a legislature with one chamber.  Unicameral Legislatures are followed by a number of countries including, Malta, New Zealand, Denmark, Sweden, Spain, Israel, and even one of our own states: Nebraska  On a national level this legislature can be seen as positive because it can make citizens feel more of a connection with their government, knowing the whole country can be represented by a single body shows that the whole country shares the same interests  Although, the more the power of government is divided up into smaller units, the safer the government would be from those who would abuse its power. 5. Executive – the branch of government responsible for putting laws into effect. 6. Electoral College – an intermediary body that elects the president.  Article II: Establishes the executive  Section 1: 4 year term, qualifications for the president [natural born citizen, 35 years old, and a resident of the US for 14 years] and the election process  Section 2: establishes the powers of the executive  Section 3: the president will periodically inform congress on how he believes the country is doing and propose the measures he thinks will best fit.  Section 4: impeachment is possible 7. Presidential System – government in which the executive is chosen independently of the legislature and the two branches are separate. 8. Parliamentary System – government in which the executive is chosen by the legislature among its members and the two branches are merged. 9. Judicial Power – the power to interpret laws and judge whether a law has been broken. 10. Judicial Review – the power of the Supreme Court to rule on the constitutionality of laws.  The judiciary was the least threatening branch of power  Has neither force nor will but merely judgment – leaves more concentration on doing what is constitutional  The court does NOT examine every law that congress passes, although with the judicial review IF a law is challenged as unconstitutional and is appealed all the way to the Supreme Court, then the justices may decide to rule on it  Because the constitution is so silent on the rules of the court, that role has been left to be defined by congress and even by the courts themselves 11. Legislative Supremacy – an alternative to judicial review, this is to merely allow the legislatures law’s to stand unchallenged and finalized. 12. Separation of Powers – not allowing the legislature, executive, and judicial branch to be exercised by the same person or groups, thereby limiting the powers of each. 13. Checks and Balances – allows each branch of government to exercise some form of control over the others  Article I: sets up the bicameral legislature  Article II: empowers the president to executive laws  Article III: creates the Supreme Court 14. Fusion of Powers – the opposite of separation of power, combining or blending branches of government.  Federalism refers to the relationship between the national government and the states  Local government is not mentioned in the constitution at all because it is completely under the jurisdiction of the states  Many powers of the national government are the powers of congress 15. Enumerated Powers of Congress – the strongest statement of national power, the list of powers congress has, followed by a clause giving congress the power to make all laws that are “necessary and proper” to carry out its power. 16. Necessary and Proper Clause – also called the “Elastic Clause”, used to justify giving congress many powers never mentioned in the constitution. 17. Supremacy Clause – of Article VI, when national and state laws conflict, the national law will be followed.  The constitution says even less about the powers granted to the states.  Although, there is the 10 amendment stating that all powers not given to the national government are reserved for the states th  Even with the 10 amendment, the necessary and proper clause make it hard for states to have the final say sometimes 18. Concurrent Powers – powers that are shared by both the federal and state governments. 19. Dual Federalism – a failed theory, the federal system under which the national and state government are responsible for separate policy areas. 20. Cooperative Federalism – unlike dual federalism, rather than being two completely separate layers, this new theory emphasized the national and state levels to be interdependent and each level required the cooperation of the other to get things done.  Federalism came to be seen by political scientists as a partnership, in which the dominant partner was, most often, the national government 21. Unitary System – the central government has all the power and the local units are dependent on it [centralized government]. 22. Confederal System – how the US was under the articles of confederation, the local units held all the power and the central government is dependent on them for its existence. 23. Federal System – the flow of power goes in both directions.  Some consequences of giving the states so much power include:  Placing states in competition with one another for scarce resources  Providing easy access to governing within a state, the more of a specific interest group there is in a state the easier it is for them to create rules and regulations, leaving those who oppose no way out  Different penalties for different crimes between states make it difficult to gauge the consequences of one’s behavior in another state 24. McCulloch v. Maryland – Supreme Court ruling of 1819, confirming the supremacy of national over state government  Whether congress had the power to charter a bank and whether Maryland had the power to tax that bank  John Marshall found it appropriate to charter a bank  Although, Maryland could not tax this bank because “the power to tax involves the power to destroy” this would mean the state could overrule the national government giving it WAY too much power 25. Gibbons v. Ogden – Supreme Court ruling of 1824, establishing national authority over interstate business  Marshall continued, deciding that New York did not have the right to create a steamboat monopoly on the Hudson River  Article I section 8 allowing congress to regulate commerce 26. Nullification – declaration by a state that a federal law be null within its borders.  The Civil War represented a giant step in the direction of a stronger national government although a big reason it was fought was to resolve the question of national v. state supremacy  Northern states were national government post civil war ending  Federalism today, most of the time people will fight to have decisions made in the arena, national or state, whether they are most likely to succeed 27. Devolution – the transfer of powers and responsibilities from the federal th government to the states [movement came to a stop following Sep. 11 ]  National government care so much about what states do because:  From a congress member’s point of view, it is easier to solve many social and economic problems at the national level  Members of congress profit electorally by passing laws and regulations that bring to their states resources  Sometimes congress prefers to adopt national legislations to preempt what states may be doing or planning to do Federal policymakers face one of the biggest challenges in this regard: how to get the states to do what federal officials have decided they should do.  Congress makes two key decisions when it attempts to influence what the states are doing: must be concerned with the character of the rules and regulations and then it must be concerned whether the cost of the new programs will be paid by the national government and how much.  With the combination of these two decisions, 4 strategies arise on how to influence states: No national government influence, categorical grants, block grants, and unfunded mandates 28. Categorical Grant – provides very detailed instructions, regulations, and compliance requirements for the states to receive federal funding for a specific purpose.  States and local governments have become so dependent on federal grants that these subsides now make up 27% of all state and local spending  These grants remain the predominant form of federal aid, 80% of all aid to state and local government 29. Block Grant – rather than detailed, combines broad program requirements and regulations with federal funding provided for a broad purpose.  The largest most significant block grant was instituted under Bill Clinton with the passage of the welfare reform act  These grants concern congress members because they fear that the state will do whatever they want instead of what congress intended  When federal funds are not attached to specific programs, congress can no longer take credit for the program 30. Unfunded Mandate – imposes specific policy requirements on the states but does not provide a way to pay for those activities.  Unfunded mandates are more attractive to members of congress in periods of ballooning national deficits  Many in the states and even the national government say they want the states to have more power or simply that all levels are government do less  Claims that “Cooperative Federalism” has been transformed in “Coercive Federalism” in which the states are pressured to adopt national solutions to their local problems with minimal state input 31. Amendability – the provision for the constitution to be changed, so as to adapt to new circumstances.  The founders provided two methods: formal and informal amendment process  Over 10,000 amendments have been introduced, but only 27 have been formally amended  In the name of interpreting the constitution, the constitution can be changed in more subtle ways by the Supreme Court, this is done without any amendments ever being passed  Article V spells out the federal procedure for officially amending the constitution, federal because they require the involvement and approval of the states as well as the national government  The method by which an amendment is proposed can affect the success of the amendment itself  Making it to easy to amend a constitution can lead to special interests push for amendments that give them tax breaks or other protection or if it’s too easy it will be done too frequent and an opinion can be unsteady  California’s constitution has been amended more than 500 times The founders wrote a constitution that profoundly limits citizen participation and the opportunities to get involved in it are few and costly in terms of time, energy, and money  The constitution was the republic’s insurance policy against chaos and instability  The constitution does an opportunity for participation, it creates a federal system, participation can flourish at the state and local levels even while it remains limited at the national level  There are three formal mechanisms of direct democracy at the state level: the initiative, the referendum, and the recall 32. Initiative – bypasses the legislature, by citizens placing a proposal or constitutional amendment on the ballot to be adopted or rejected by majority vote. 33. Referendum – An election in which a bill passed by the state legislature is submitted to voters for approval 34. Recall Elections – votes to remove elected officials from office Chapter 5: Fundamental American Liberties 1. Civil Liberties – protect our rights to think and act without government interference 2. Civil Rights – secure citizenship rights to all members of society, the government must treat all citizens equally [primarily stated in the 13, 14, 15, 19, and 26 amendment]  Although one person’s freedom to speak or act in a certain way may be limited by another persons rights  Non authoritarian governments give citizens the power to challenge government if they believe it has denied their basic rights, unlike authoritarian government  Civil Liberties provide rules that keep government limited so that it cannot become too powerful  While, Civil Rights help define who “we the people” are in a democracy and they give those people the power necessary to put some controls on the government  Until the writing of the Enlightenment, it was rare for individuals to talk about claiming rights against government  John Locke argued that the primary purpose of government was to preserve the natural rights of its citizens  To claim that an individual has rights that must be respected by government is a citizen  While, a person under authority of a government but cannot claim rights is merely a subject  People clash over rights in two ways: conflict occurs between individuals [praying during school] and when the rights of individuals are pitted against the needs of society and the demands of collective living [wearing a helmet while riding a motorcycle is a law]  Measures to increase security of US citizens also reduce their civil liberties  Courts agree to hear cases when it wants to send a message to lower courts about how the constitution should be interpreted  While, congress has chosen to stay out of disputes about rights although at other times it will take action to either limit or expand rights  The president gets involved by having administration officials lobby the supreme court to encourage outcomes they favor  Individual Americans may use the courts to sue for what they perceive as their rights but more often individuals act in groups [ACLU] 3. Habeas Corpus – the right of an accused person to be brought before a judge and informed of the charges and evidence against him. 4. Bills of Attainder – no type of government can pass, in which specific person or groups are detained and sentenced without trial. 5. Ex Post Facto Laws – no type of government can pass, in which laws make an action a crime after it happened, even though it was legal when it happened. 6. Incorporation – supreme courts process of incorporating MOST of the protections of the Bill of Rights into the states 14 amendment. 7. Selethive Incorporation – not all of the amendments could be incorporated in the 14 amendment so with the selective incorporation the Supreme Court had to consider each right on a case­by­case basis to see how fundamental it was. 8. Establishment Clause – the first amendment guarantee that the government will not create and support an official state church. 9. Separationists – supporters of a “wall of separation” between church and state. 10. Accommodationists – supporter that the states should not be separate from religion but rather should accommodate it, without showing preference. 11. Lemon Test – 3 pronged rule used by the courts to determine whether the establishment clause is violated. 12. Free Exercise Clause – the first amendment that citizens may freely engage in the religious activities of their choice. 13. Police Power – the ability of the government to protect its citizens and maintain social order. 14. Compelling State Interest – a state must show that it is absolutely necessary for some fundamental state purpose that the religious freedom be limited. 15. Sedition – a restrictive legislation, speech that criticizes the government to promote rebellion. 16. Bad Tendency Test – if congress has the right to outlaw certain actions, it also has the right to outlaw speech that is likely to lead to those actions. 17. Clear and Present Danger Test – allows language to be regulated only if it presents an immediate and urgent danger. 18. Imminent Lawless Action Test – restricts speech only if it is aimed at producing or is likely to produce imminent lawless action. 19. Freedom of Assembly – the right of the people to gather peacefully and to petition government. 20. Miller Test – rule used by the courts, in which the definition of obscenity [wrongful words] must be based on the state or local level/standards. 21. Fighting Words – speech intended to incite violence. 22. Political Correctness – the idea that language shapes behavior and therefore should be regulated to control its social effect. 23. Prior Restraint – a restriction on the press BEFORE its message is actually published [very dangerous because it caused a censorship on something that never entered the public and their worth could not be debated]. 24. Libel – the written slander [defamation] of character. 25. Due Process of Law – laws will be fair and reasonable and those who break the law will be treated fairly. 26. Exclusionary Rule – rule created by the Supreme Court, stating that evidence seized illegally may not be used to obtain a conviction.

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Chapter 5.3, Problem 11E is Solved
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Textbook: Fundamentals of Differential Equations
Edition: 8
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
ISBN: 9780321747730

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Solution: In 10–13, use the vectorized Euler method with h =