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Differential Equations, part 1: Elimination of Arbitrary Constants
1. At the end of this hub is paragraph 10, going to heaven . . . la la la; par. 11, a study axiom; par.13, something to ponder.
THE NEXT LEVEL:
1.1 At the end of this hub is a link that will take you to the next level of eliminating arbitrary constants to arrive at a general differential equation.
WHAT IS A DIFFERENTIAL EQUATION?
2. You can look it up in a dictionary which will tell you it is "An equation containing derivatives or differentials of an unknown function." When I think of a differential equation( DE) I think of a snapshot; a picture freezing time. There may be more than one snapshot, but whether it is one or more, we can use the techniques developed to solve DEs and determine the function that is behind those snapshots. These snapshots( derivatives) are described at paragraph 2 of hub#12.15. The mathematics of DEs is rich, deep and exciting; it is amazing! The power of DEs is enormous, enabling engineers and scientists to solve numerous and difficult problems. The many techniques developed to solve DEs are ingenious. If you love mathematics, then you will be in math heaven with differential equations.
WHERE AM I GOING TO START?
3. My math hubs don't start at the beginning. The main criterion for doing a math hub is, is it interesting? I think finding the logarithm with a negative argument(linked) is very interesting. They are not really designed to be very tutorial. Their purpose is to generate a fascination and appreciation for math. I have not done a hub on algebra yet--actually I did do one during the time I was working on this hub--but when I do it probably will not start at the beginning as with variables, domain, range, functions---BORING! It will probably be on partial fractions because partial fractions are interesting. Fortunately my hubs on differential equations( DEs) will closely follow what is chronologically taught in a classroom. The reason is simple: There is absolutely noting boring about DEs. Every aspect of them from beginning to end is fascinating. So I'll start at the beginning, The Elimination of Arbitrary Constants.
WHAT HAPPENS WHEN THE CONSTANTS ARE ELIMINATED?
4. Solving a differential equation( DE) is finding the function upon which the DE is built. That would be Y at L1. Elimination of the arbitrary constants is working backward to find the DE, which in this case is at L11, and also at L19. L11 is a second order, homogeneous, ordinary differential equation( DE), as explained at paragraph 4 of hub#12.16( linked). So when you eliminate the arbitrary constants of the solution( Y at L1) to a DE then you get the DE itself at L11. All the skills you learned in basic math will be applied when working with DEs.
L1 THROUGH L11 EXPLAINED:
5. The number of times we must differentiate an original function( L1) depends on how many arbitrary constants there are. We have two( C1 and C2); therefore, we must differentiate Y at L1 twice in order to eliminate both. As explained at hub#12.17( linked), differentiating an exponential function to the base e is unbelievably easy. The exponent is a function of x ; therefore, the chain rule is applied giving us the factors -2, and 3 at L3, and the factors 4, and 9 at L3. The chain rule is used with ex also but since the derivative of x is 1, then the value of ex does not change when it is differentiated. I explain the chain rule at paragraph 4 of hub#12.17(linked), and a link to a hubber giving some more examples is just below that paragraph. We make two simultaneous equations from L2 and L3, and add them at L4, which eliminates C1 . We convert L5 into L7 by multiplying by 3 at L6. At L8 we multiplied L2 by 2, and added the result to L3, and their sum gave the answer at L9. Finally at L10 we subtract L7 from L9 to eliminate C2 , and we are left with the DE at L11 with no constants( C1 and C2 ).
MATRIX SOLUTION IS MUCH EASIER:
6. Solving this same problem with matrices is much easier. The only thing I really needed to do was use the right matrix of L14; the left matrix is not needed, and neither is L13. Just solve the determinant of the right matrix at L14, and that is your answer at L19. We'll go through the steps anyway.
L13 THROUGH L15, AND COFACTOR SIGN EXPLAINED:
7. Notice if we subtract Y from both sides of the equation at L1 we get row 1 at L13. We do the same thing for L2 and L3 for rows 2, and 3 of L13. We now have 3 simultaneous equations set up at L13. We write these in matrix form at L14. Next we find the determinant of L14. Finding the determinate of a 4 X 4 matrix was done at hub#12.2( linked). We could have found the determinant of that matrix by expansion of minors, but it is a pain to do it that way for anything over a 3 X 3 matrix. This method( expansion by minors) is explained well by calculus-geometry( linked to that hub). We will use this method to find the determinant of the matrix at L14. At his hub he mentions alternating the sign of each factor that is multiplied with its corresponding minor matrix. What I did at L15 is include the formula that determines that sign. It comes in handy for very large matrices. The (-1)1+1 factor at L15 is to determine the sign of the leading factor, -Y. The 1+1 exponent is the first row, and first column added( row 1 plus column 1). If that sum is even( as in this case since 1 + 1 = 2) then this cofactor is -1 times -1, which gives us a positive 1; therefore, the sign is positive. A positive 1 times -Y( negative Y) gives us a -Y for our leading factor. The second leading factor is a2,1 , which is -Y'. Since it is from row 2, and column 1 we have 2+1=3 for the exponent of the -1, which gives a cofactor of -1, making the leading factor Y' since -Y' times -1 gives a positive Y' as our leading factor. For the 3rd line of L16 the exponent is even; therefore, the cofactor is a positive 1 making the leading factor here to be -Y''. It is much easier to just alternate the signs as calculus-geometry shows us in the above link. My way is the formal way to do it and it helps with large matrices, and if you do not remember to start with a negative or positive when you start alternating signs.
COMMON TERMS AT L15 SIMPLIFY MATRIX AT L14:
8. So we will use column 1 for our leading factors( -Y, -Y', -Y'') and cofactors( -1row + column)) to multiply by the determinants of their respective minor matrices. Calculus-geometry has a hub( linked) on this also: finding the determinant of a 2 X 2 matrix. I also mention it after paragraph 13 at LE1 to LE10( linked to that hub). At L15 we follow the instructions at the previous two links, and notice we have common factors. This is true for all the minor matrices being expanded by column 1. This determinant is set to zero. Since we have these common factors in the entire expansion process, and since the determinant is set to zero, then we can factor out the common factors, and divide both sides of the equation by the common factors to get rid of them. That leaves us with the matrix on the right of L14, which is much simpler.
L16 THROUGH L19 EXPLAINED:
9. So we now just find the determinant of the 3 X 3 matrix at the right of L14; the link at paragraph 7 will walk us through this. We have the leading factors, their cofactors( giving the proper sign), and the determinants of the minor matrices. We go through the arithmetic and get an answer at L18. The determinant was set to zero, so our equation is equal to zero; therefore, we factor out a --5( negative five) from L18 to get rid of it, and then set the equation in order of highest to lowest derivative, and the final answer is at L19, which is the same as L11.
LET'S ADD A VARIABLE TERM TO THE SAME EXAMPLE:
E1: At L20 a variable term was added to the same example; i.e. added to L1. Nothing has changed as far as what we do. We still differentiate twice to eliminate the two arbitrary constants, and we still use algebraic manipulations to eliminate the terms within which the arbitrary constants exist. So we take the first and second derivatives at L20 through L22. Then at L23 and L24 we eliminate C1 between eq9 and eq10. We do it again for C1 between eq10 and eq11, and that is done at L26 and 27. Next we want to eliminate C2 from the equations at L24 and L27. This is done at L28 by multiplying L24 by a negative 3( i.e. a -3), and adding this to L27 thereby eliminating the C2 terms. At L29 we have an equation with no arbitrary constants, and at L30 is the final form.
EVERYONE WILL BE SAVED AND GO TO HEAVEN . . . UH-HUH:
10. Actually, that one statement, everyone will be saved, encompasses all three statements at paragraph 10 of hub#12.16( linked). God's love would have to be unconditional, and everyone would have to be God's children to make the first statement true: everyone is saved. Paragraph 10 begins with the fact that we are being buried in sophistry. Sophistry is not just deception, but it is clever deception, and in these last days it will be extremely clever with the apparent credentials to back it up( Matthew 24:24). An example of a flat out lie is when the serpent contradicted God by saying, "You shall not surely die"( Genesis 3:4), after God just got done saying, "you shall surely die"( Genesis 2:17). However, the serpent also put a bit of sophistry in the mix with "you shall be as gods, knowing good and evil,"( Genesis 3:5) because what comes to mind is gods do not die, and you must be a god because you know everything. So that hubber, along with millions of others, have a similar way of contradicting God. They believe, and claim to others, that everyone will be saved when in fact God has made it clear that is not true( Matthew 22:13,14; Mark 9:42--47). But what about 1Timothy 4:10? Having Scripture to back up one's opinion is a powerful argument. Scripture never contradicts itself. That is something all Christians, and want-to-be Christians must realize, but we definitely have an apparent contradiction here.
AN AXIOM OF BIBLE STUDY:
11. An axiom of Bible study is there are no contradictions in Scripture, no not in God's Word. John MacArthur in his exemplary study Bible( The MacArthur Study Bible) goes into considerable detail explaining this verse( 1 Timothy 4:10) when it seems to contradict Matthew 25:41,46; Revelation 20:11-15; and the Scripture I gave above. He says, "God is the Savior of all men, only in a temporal sense, while of believers in an eternal sense." He goes on to develop--backed with much Scripture--how this is the case in explaining 1Tim. 4:10. An example of what he mentions is Matthew 5:45. God maintains the rain, food, weather--well, the entire biosphere, and everyone benefits: good and evil, believers and non-believers.
12. I doubt it will be the next math hub, but when I continue on this subject( par. 10 of hub#12.16), I'll link you to it from here.
ONE FINAL THOUGHT CONCERNING WISHFUL THINKING:
13. As paragraph 12 of hub#12.16(linked) states, God's Word is the Standard, not our worthless opinions( Isaiah 5:20). Wishful thinking, based upon erroneous opinions, will not deliver us from the relentless and deceptive tentacles of hell to the joy of heaven. The decision concerning Jesus Christ is paramount, and the Way to salvation is specifically described in the Bible. Saving faith is obeying God, stopping sin--turning and walking away from sin. This new age belief that everyone is saved is wishful thinking to dupe one into believing he/she can continue to sin and still get to heaven; i.e.,not go to hell; it is one or the other. Ponder this. You are alive now, and you will always be alive . . . forever. These funerals we go to do not represent the ending of a life. They represent the ending of second chances. Do not buy into the sophistry that we can continue to sin when Scripture is so clear that it has got to stop( 1John 3:7,8,9,10). YOU DO NOT WANT TO GO TO THE BAD PLACE!
Elimination of arbitrary constants with a single variable as two factors:
- Differential Equations, Part 3: Elimination of Arbitrary Constants with a Single Variable in Two Fac
The variable as two factors adds complexity but it can be handled by equating the elements of the vectors to zero at L10. A fundamental theorem of DE's makes it even easier.