Differential Equations, Part 2; The Differential Operator:
Sophistry, The Standard, and belief based on sound reasoning are mentioned at paragraphs 10, 11, 12 and 13.
WHAT IS THE DIFFERENTIAL OPERATOR?
1. My math dictionary says an operator is the same as a function. An operator mathematically operates; i.e. it tells you what to do. In that sense the +, -, x, / (plus, minus, times, divide) are operators. The line equation, ax + b is an operator. So if a = 3, and b = 4, then we have 3x + 4. Given x = 7, this operator gives us 3(7) +4 = 25. It tells you what to do with the variable x, and in this case it maps x into 25.
2. The differential operator, D, tells you to differentiate. Let's say Y = x7, then DY = 7x6, and D2Y = 42x5, and D3Y = 210x4, etc. So the superscript of D tells you how many times to differentiate. If Y is any function that is differentiable, then D can operate( i.e. differentiate in this case) on it.
AN AMAZING AND POWERFUL TECHNIQUE:
3. L1 and L4 do not express my confusion concerning the differential operator itself, but rather what can be done with it--it has enormous power in solving many DEs( differential equations). What one does is rewrite the DE, as at L2 and L5, in terms of the DO( differential operator), then solve the roots of the DO equation, and these roots give the solution to the DE. . . . WHAT? Yeah, in the TWILIGHT ZONE maybe, but not in this world! How can this possibly be? So I can just willy nilly write the DE as a DO equation, solve the zeros, and that is my solution. That is RIDICULOUS! These thoughts are what went through my mind when I first studied this technique. Even if the roots are complex, they will represent solutions. OK, fine, I went ahead and did it because I got the right answers, but it really bugs me when I'm using a mathematical technique, and I do not have at least an intuitive reason why it works. This hub gives you that intuitive reason.
L1 THROUGH L7 EXPLAINED:
4. Paragraph 3 introduced L1 through L7, but let's take a closer look. L2 represents a first order( which means the highest-ordered derivative), homogeneous( which means the equation equals zero instead of something else), ordinary( which means the derivatives are total derivatives, not partial derivatives), differential equation( DE). L5 represents a second order-homogeneous-ordinary DE. We assume the solution for L2 involves the base of the natural logarithms( linked) as at L3, but we do not know the value of m1 at L3 if not for the solution of D at L2. The same goes for L5. We assume the solution involves "e" as shown at L7, and the answer( solution for the DE) is also given at L7 because we solved the roots of the quadratic( at L5) at L6 which gives the values of m1 and m2 , which we plugged in at L7.
THE DIFFERENTIAL OPERATOR( L8 THROUGH L12.1):
5. The differential operator( DO) does what the familiar operators listed at L9 and L10 do: differentiate functions. L11 is the DO of order "n", and an are the constant coefficients; however, the an's can also represent functions, not just constants. When Y is distributed across the DO as at L13 then we get the form of L12 and 12.1 in which the various nth order derivatives are executed upon the function Y = f( x).
WHY THE DIFFERENTIAL OPERATOR WORKS TO SOLVE DIFFERENTIAL EQUATIONS( L13 THROUGH L18):
6. At L13 we write the DE in the form of the DO, and we assume the solution to the DE has the form represented at L14. The solution to a DE is finding the original function, Y = f( x). We need to find the values of m1 , and m2 at L14. C1 and C2 are no sweat because they just represent a family of curves that behave as the function Y = f( x) dictates. We only distribute one term of L14 through the DO because we do not need the other term. Either term will give us the correct values of m1 and m2 . Y at L14 is a solution to our DE, but each term of Y is also a solution( these are called particular solutions). So we will use one of the particular solutions in Y to solve for all of Y( i.e. finding the values of m1 and m2 ). When we distribute Y through the DO equation at L13, we multiply Y by the constant, 12; however, because Y is to the right of the DO we operate on Y; we do not multiply it. To the right of D2 we take the 2nd derivative of Y, and to the right of D we take the 1st derivative, and to the right of 12 we just multiply it. This is all set up at L15, and 15.1, and the DO is executed at L16. You will be completely lost if you do not know how the differentiate. Village people( linked) gives a thorough description of what is differentiation, and Sierranicole24 gives( linked) several rules of differentiation. The first one, the Power Rule, is applicable to this hub.
7. Incidentally, the last factor of L15.1 was divided out on both sides of the equation so we could get rid of it. As explained previously we do not need C2em2x to find the values of m1 and m2 . The Power Rule( see last link above) was used at L16, and I describe how to differentiate an exponential function to the base e at hub#12.17( linked). Notice we have a common factor at L16; therefore, we can factor out that common factor and divide both sides of the equation at L16 with that factor to get rid of it, and that leaves us with L17. Notice it is the same quadratic as our DO equation at L5. So we can solve for D or m1 in either equation and we get L18, which gives the roots( zeros) at L19.
CHECK TO MAKE SURE WE HAVE THE CORRECT ANSWER:
8. We plug the values of L19 into our assumed answer( see L3) at L20, and we work through the arithmetic from L21 through L24, and it does indeed equal zero.
9. This is a remarkable result that the DO can so greatly facilitate the solutions of so many DEs. As I mentioned previously, even if the roots are complex we can use Euler's formulas( mentioned at hub#12.12( linked)), and find solutions to the DEs.
SOPHISTRY--WE ARE GETTING BURIED IN IT:
10. There seems to be three beliefs that continue to gain ground: 1st. Everyone will be saved( go to heaven)--1 Timothy 4:10; 2nd. Perfect love is unconditional love; 3rd. All people, regardless of beliefs and religions, are God's children--Ephesians 4:6. I will address each one of these at the end of my math hubs, and you will be linked to each argument against them, beginning with this hub, when each argument is completed. The Scripture given in this paragraph was listed by a hubber who believes in all three. These three beliefs are the antithesis of what the Bible teaches, but they are only three of copious beliefs that contradict God's Word. Fallacious beliefs and statements can have Scripture attached to them as Satan did with Christ( Matthew 4:6), but one must use all of Scripture to determine what is true or false as Christ did with the Satan(Matt. 4:7).
11. It is remarkable how prevalent is sophistry throughout the world, let alone America. It is used to justify socialism, homosexuality, abortion, tolerance when rebuke is required, inflated wages, lying, ridiculous demands by unions, hatred of companies and the rich, injustice in many forms, cruelty to animals; rape of the taxpayer, especially the rich; and nearly everything else that is not congruent with Scripture.
GOD'S WORD--THE STANDARD:
12. Sometimes sophistry can be so well put, and so effective that the only way to identify it is with God's Word, the Bible. For a Christian, or even a professing christian, the Bible is the Standard by which all thoughts, actions, beliefs, arguments, behaviors, reasons, and logic are to be evaluated and judged, and then confirmed, or disregarded. More detail concerning sophistry is at paragraph 13 of hub#12.12( linked). The three beliefs at paragraph 10 above imply that regardless of your beliefs, religion, ongoing sins, lack of works, lukewarmness to Christ, you are saved; everyone is saved! It really makes no difference what you do or don't do. Yet paragraph 13 of hub#12.9( linked) has the Scripture that confirms we must obey God through Jesus Christ if we expect to get to heaven. God's Word is the Standard; we must trust that it means what it says in its entirety. Many people, including professing christians, believe hell does not exist, and some churches and religions embrace this belief. God's Word, the Bible, is the Standard to determine the reality of eternal hell. Paragraph 9 of hub#12.10( linked) has the Scripture to confirm that fact, and the conditions to stay out of eternal torment. Paragraph 10 of that hub mentions that our arbitrary opinions can lead us to hell. Do we actually think our opinions are a formidable force against God, His Word, Jesus Christ, and the Holy Spirit?
WHY I BELIEVE IN GOD, THE BIBLE, JESUS CHRIST AND THE HOLY SPIRIT:
13. The evidences! It is--to me--a no brainer. Paragraph 8 of hub#12.14( linked) gives an analogy of why random processes cannot build anything of significance, let alone structures as eyes, ears and brains that depend on other structures as gastrointestinal system, circulatory system and heart. Not only do the evidences within creation prove that God exists, but they also testify to God's omniscience and omnipotence. What about Jesus? How many more credentials does one need to prove He is who He says He is than healing hundreds, if not thousands of people, controlling the weather, walking on water, and raising the dead? The Bible is clearly a supernatural Word from God because of the fulfilled prophesies. Hub# 2 backs that up mathematically( linked). There is no way such consistently accurate predictions of the future can be made if not for the One Who inhabits eternity( Isaiah 57:15). The Bible speaks of the Holy Spirit, and that is all the proof we need to believe in Him, and anything else the Bible mentions.