@@Ajax2696 THE PRECISE, SIMPLE, TOP DOWN, AND CLEAR DERIVATION, ORIGIN, MEANING, AND CONSTRUCTION OF E=MC2: E=mc2 is taken directly from F=ma. BALANCED inertia/INERTIAL RESISTANCE is fundamental. Carefully consider what is THE SUN AND what is the speed of light (c) ON BALANCE. The stars AND PLANETS are POINTS in the NIGHT SKY. So, consider what is THE BLACK “space” of what is THE EYE. The sky is blue, AND what is THE EARTH is ALSO BLUE !! c squared is CLEARLY understood as a dimension (of what constitutes SPACE) ON BALANCE, as ELECTROMAGNETISM/energy is gravity ON/IN BALANCE !!! GREAT !!! I have CLEARLY explained or proven the mathematical unification of gravity AND ELECTROMAGNETISM/energy (ON/IN BALANCE), as I have demonstrated the fourth dimension AND the term c4 from Einstein's field equations (along with TIME); as ELECTROMAGNETISM/energy is gravity (ON/IN BALANCE). I have proven how and why this is mathematically consistent with F=ma AND E=mc2. TIME dilation ultimately proves on balance that ELECTROMAGNETISM/energy is gravity (ON/IN BALANCE). In light of what has been CLEARLY proven here (on BALANCE), think about what is THE SUN in DIRECT comparison to/with what is outer “space”. GREAT !!! It ALL CLEARLY makes perfect sense, as BALANCE and completeness go hand in hand. I'm going to more precisely explain the true origin (and the full meaning) of the equation E=mc2. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE, as ELECTROMAGNETISM/energy is (CLEARLY and necessarily) proven to be gravity (ON/IN BALANCE). This also CLEARLY explains, on balance, why and how this equation represents a two dimensional surface OR SPACE as what is a BALANCED AVERAGE. Indeed, gravity AND ELECTROMAGNETISM/energy are (CLEARLY) linked AND BALANCED opposites (on balance); as ELECTROMAGNETISM/energy is gravity (on/in balance) !!! Energy has/involves GRAVITY, AND ENERGY has/involves inertia/INERTIAL RESISTANCE. TIME dilation ULTIMATELY proves ON BALANCE that ELECTROMAGNETISM/energy is gravity. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE. Indeed, TIME is necessarily possible/potential AND actual IN BALANCE; as ELECTROMAGNETISM/energy is gravity (on/in balance). Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE on balance, as ELECTROMAGNETISM/energy is gravity ON/IN BALANCE. This explains F=ma AND E=mc2. This ALSO explains why the rotation of WHAT IS THE MOON matches it's revolution. Inertia/INERTIAL RESISTANCE is proportional to (or BALANCED with/as) GRAVITATIONAL force/ENERGY, as ELECTROMAGNETISM/energy is gravity (ON/IN BALANCE). This explains F=ma AND E=mc2. I have truly (and fundamentally) revolutionized what is our understanding of physics/physical experience. Compare the setting and orange Sun with the fully illuminated and setting/WHITE Moon DIRECTLY. They do basically appear to give off the same illumination, in fact. Notice that the curvature or shape of said Moon matches that of THE EARTH/ground (given a clear horizon, that is). What is THE SUN AND what is THE MOON are then the SAME SIZE in the sky as what is THE EYE. Moreover, these two forms manifest (or form up) at what is EYE LEVEL/body height. GREAT !!! Now, I have CLEARLY proven why AND how this is so. The tides are (CLEARLY AND NECESSARILY) ELECTROMAGNETIC/gravitational ON BALANCE. Magnificent !!! The tides are CLEARLY and necessarily subject to F=ma AND E=mc2, as ELECTROMAGNETISM/energy is gravity (ON/IN BALANCE). Again, the stars AND PLANETS are POINTS in the night sky; and the rotation of WHAT IS THE MOON matches it's revolution. The sky is blue, and THE EARTH is ALSO BLUE. Excellent. I have explained why the sizes of said Moon and said Sun are the same as what is THE EYE ON BALANCE. So, I have CLEARLY explained why the diameter of the Moon is about one quarter (at 27 percent) in size compared with what is THE EARTH. How does what is the Sun survive or exist against what is outer “space" ? The answer is, ON BALANCE, invisible AND VISIBLE SPACE in fundamental equilibrium and BALANCE. Here's why. (Think about TIME as well.) WHAT IS THE EXPLANATION OF SPACE AND TIME ON BALANCE: Invisible AND VISIBLE SPACE in fundamental equilibrium AND BALANCE. This necessarily and clearly involves interaction, on balance. Consider what is the eye (on balance). Logically consider what is a two dimensional surface OR SPACE ON BALANCE !!!! Notice the associated DOME AND the flat/black “space” of WHAT IS the eye AS WELL. Really think about it all. Consider WHAT IS THE SUN ON BALANCE. (NOW, think about time.) Outer “space” involves full inertia, AND it is fully invisible AND black. Consider one and three dimensions ON BALANCE !!! Now, consider what is the fourth dimension and the term c4 from Einstein's field equations. Think about ELECTROMAGNETISM/energy AND think about gravity (ON/IN BALANCE). Consider what is the man (AND THE EYE ON BALANCE) who IS actually standing ON what is THE EARTH/ground (ON BALANCE) !! Think about TIME !! Think about why there is something instead of nothing ON BALANCE. Consider that time is NECESSARILY possible/potential AND actual ON/IN BALANCE. Think about the man (THE EYE) that actually IS IN what is outer “space”. Think about time. The stars AND PLANETS are POINTS in the night sky ON BALANCE, and consider what is the speed of light (c) ON BALANCE. NOW, consider what is THE SUN. Think closely about everything in this writing. Balance and completeness go hand in hand. Magnificent. ❤️ c squared CLEARLY means an INTERACTION on balance, as the stars AND PLANETS are POINTS in the night sky. Balanced inertia/INERTIAL RESISTANCE is fundamental. Consider E and “m" on balance. Great. The fourth dimension is only consistent with what is (on balance) a TWO dimensional surface OR SPACE ON BALANCE. Consider what is the eye. Consider what is the balanced MIDDLE DISTANCE in/of SPACE (ON/IN BALANCE). Consider invisible AND VISIBLE SPACE in fundamental equilibrium AND BALANCE. The stars AND PLANETS are POINTS in the night sky ON BALANCE. So, consider what is the speed of light (c) ON BALANCE. This CLEARLY explains the fourth dimension AND the term c4 from Einstein's field equations. Time is NECESSARILY possible/potential AND actual ON/IN BALANCE, as BALANCED inertia/INERTIAL RESISTANCE is fundamental pursuant to F=ma AND E=mc2 in balance. A galaxy consists of invisible AND VISIBLE SPACE in fundamental equilibrium and BALANCE, thereby eliminating the need for any "dark" "matter" or "dark" "energy"; as ELECTROMAGNETISM/energy is then CLEARLY gravity ON/in BALANCE. By Frank DiMeglio
Ok genius iam also approaching the problem same way like you I don't use matheMatical way my question is so simple because LTI depends on convulution here's my question below Convulution is nothing but stacks and scales the input that's why the input to an amplifier is stacked and scaled or amplified but in filter design it attenuate frequency so I don't know how it regret certain frequency by stacking and scaling the input if possible some one explain to me Also application Brother I know mechanical engineers could find resonance but when I had a deep thought on this resonance Is an slow accumulation of energy which is accumulated very high in small installments when the frequencies match if you strike a turning fork of 50 hz you get the same frequency of vibration on another tuning fork so they both vibrate if you strike it harder the amplitude changes hence loudness is a human factor the frequency is the same the languages that human speak through out the world the sound only resonate your ear drum for few seconds my question is that the harmonics is the fundamental frequency and overtones are the frequency that follow it take a word in any language you spell it according to convolution the thing scales and ques and stack the signal so convolution can be used to model resonance so when your ear drum vibrates it vibrates so the electrical signals are carried to brain like tuning fork ear drums vibrate within the audible spectrum 20 hz to 20000 hz hence resonance is caused by the word we speak and within the audible range the ear drums vibrate and we make sense of words I have seen in one videos on TH-cam that due to harmonics in any sound causes resonance which could be modelled by convolution recalling the resonance its destructive because slow and steady accumulation of sound on the mass causes high stress and high energy to build inside and stress increase and the system fractures or collapses but our ear drum hearing the sound from human languages try to vibrate but why our ear drum when subjected to continuous exposure of sound does not fracture or rupture like a wine glass iam not telling about high loud sound higher than 80 db but a audible range sound within the frequency of 20 hz to 20000 hz under continuous exposure why it's not damaging it again not failure by high energy but low one in synchronisation on air . But I tried it in my students when I told them to be quite in class they did not listen to me so I took my phone and set an frequency 14000 hz and they told it was irritating the idea of resonance is "small effort but large destruction " just like Tacoma bridge where the wind just slowly accumulated energy on the bridge and it collapsed it so my conclusion is if an audible frequency at continuous exposure to an human ear can it cause bleeding again "small effort but great destruction" sorry for the long story I you are able to reach hear you must be as curious as me so still not finished the ear drum is shook by harmonics in the sound we make by the words( or )overtones in the sound we make by the words I know harmonics is the fundamental frequency and overtones are following it which under slow and steady accumulation of sound energy resonates and could damge the ear drums again "small effort but big destruction" not to mention we assume the person is in coma or brain dead hence when the sound irritates him he or she could not make a move so my question is so simple normally human ear responds to harmonics or overtones according to convolution which could be a disaster but with minimal effort 🙏🙏🙏🙏 at here I could be wrong because harmonics can also be used to construct sound so can it be destructive or the overtones which are the trouble makers and which one according to this gives a response curve when two signals convolved by harmonics or overtones which is destructive but with minimal effort and convolution happens when ear drums oscillate is by harmonics or the overtones or also the trouble makers there
has no application to this situation whatsoever, destructive and constructive effects don't apply at all in real world scales, go find out what the meaning of harmonics and overtone is and how u define pain is absurd too, implying pain to damage is bullshit
This video series develops those subjects both separately and together and supplements Prof. Strang's textbook, Differential Equations and Linear Algebra: www-math.mit.edu/~gs/dela/
It's Prof. Strang's book: Differential Equations and Linear Algebra www-math.mit.edu/~gs/dela/, you can check out the complete OCW course site for more information: ocw.mit.edu/RES-18-009F15.
Often times, the point is to use the Laplace transform, to avoid the need to do the convolution integral. A typical example where it may seem that the Laplace transform cannot work with partial fractions, is convolving combinations of either sine with itself, cosine with itself, or the two with each other. Partial fractions directly do not help you, but there is an alternative method to still work in the Laplace domain and solve it. Consider sin(t) conv sin(t). This sets up the Laplace transform of 1/(s^2 + 1)^2. You could do it by convolution integral using many trig identities, or you could do it by anticipating the form of the solution in the time domain, and matching coefficients to the corresponding Laplace domain. Assume the solution is a linear combination of sin(t), cos(t), t*sin(t), and t*cos(t): A*sin(t) + B*cos(t) + C*t*sin(t) + D*t*cos(t) We know £{sin(t)} and £{cos(t)} from a reference table, but need the s-derivative property to find it for the remaining two functions: £{t*f(t)} = -d/ds £{f(t)} £{t*sin(t)} = -d/ds 1/(s^2 + 1) = 2*s/(s^2 + 1)^2 £{t*cos(t)} = -d/ds s/(s^2 + 1) = (s^2 - 1)/(s^2 + 1)^2 Thus: 1/(s^2 + 1)^2 = A/(s^2 + 1) + B*s/(s^2 + 1) + C*2*s/(s^2 + 1)^2 + D*(s^2 - 1)/(s^2 + 1)^2 Clear fractions, expand, and gather: 1 = A*(s^2 + 1) + B*s*(s^2 + 1) + C*2*s + D*(s^2 - 1) 1 = A*s^2 + A + B*s^3 + B*s + C*2*s + D*s^2 - D 1 = (A + D)*s^2 + B*s^3 + (B + 2*C)*s + (A - D) Equate corresponding coefs: A + D = 0; B = 0; B + 2*C = 0; A - D = 1 Solution for A, B, C, & D: A = 1/2, B=0, C=0, D = -1/2 Thus, the solution is: 1/2*sin(t) + 1/2*t*cos(t)
I wish someone had taught us so well in the University. Hats off this is gem.
Dr. Strang’s texts are also terrific.
Gilbert Strang, perfect as usual.
No words can describe the beauty of this teaching, I wish I had the possibility in the past to study like this
i love this man, he makes me happy
"because I don't plan to check" had me laughing out loud in the library, great video. thanks.
Respect sir! You're great!
This is a solid introduction to convolution.
a very good introduction
Me: Education is a scam
The internet: hold my beer *plays this wonderful video :)*
Thank you!
You know I was having the same train of thought weird isn't it?
@@Ajax2696 THE PRECISE, SIMPLE, TOP DOWN, AND CLEAR DERIVATION, ORIGIN, MEANING, AND CONSTRUCTION OF E=MC2:
E=mc2 is taken directly from F=ma. BALANCED inertia/INERTIAL RESISTANCE is fundamental. Carefully consider what is THE SUN AND what is the speed of light (c) ON BALANCE. The stars AND PLANETS are POINTS in the NIGHT SKY. So, consider what is THE BLACK “space” of what is THE EYE. The sky is blue, AND what is THE EARTH is ALSO BLUE !! c squared is CLEARLY understood as a dimension (of what constitutes SPACE) ON BALANCE, as ELECTROMAGNETISM/energy is gravity ON/IN BALANCE !!! GREAT !!! I have CLEARLY explained or proven the mathematical unification of gravity AND ELECTROMAGNETISM/energy (ON/IN BALANCE), as I have demonstrated the fourth dimension AND the term c4 from Einstein's field equations (along with TIME); as ELECTROMAGNETISM/energy is gravity (ON/IN BALANCE). I have proven how and why this is mathematically consistent with F=ma AND E=mc2. TIME dilation ultimately proves on balance that ELECTROMAGNETISM/energy is gravity (ON/IN BALANCE). In light of what has been CLEARLY proven here (on BALANCE), think about what is THE SUN in DIRECT comparison to/with what is outer “space”. GREAT !!! It ALL CLEARLY makes perfect sense, as BALANCE and completeness go hand in hand.
I'm going to more precisely explain the true origin (and the full meaning) of the equation E=mc2. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE, as ELECTROMAGNETISM/energy is (CLEARLY and necessarily) proven to be gravity (ON/IN BALANCE). This also CLEARLY explains, on balance, why and how this equation represents a two dimensional surface OR SPACE as what is a BALANCED AVERAGE. Indeed, gravity AND ELECTROMAGNETISM/energy are (CLEARLY) linked AND BALANCED opposites (on balance); as ELECTROMAGNETISM/energy is gravity (on/in balance) !!! Energy has/involves GRAVITY, AND ENERGY has/involves inertia/INERTIAL RESISTANCE.
TIME dilation ULTIMATELY proves ON BALANCE that ELECTROMAGNETISM/energy is gravity. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE. Indeed, TIME is necessarily possible/potential AND actual IN BALANCE; as ELECTROMAGNETISM/energy is gravity (on/in balance).
Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE on balance, as ELECTROMAGNETISM/energy is gravity ON/IN BALANCE. This explains F=ma AND E=mc2. This ALSO explains why the rotation of WHAT IS THE MOON matches it's revolution. Inertia/INERTIAL RESISTANCE is proportional to (or BALANCED with/as) GRAVITATIONAL force/ENERGY, as ELECTROMAGNETISM/energy is gravity (ON/IN BALANCE). This explains F=ma AND E=mc2. I have truly (and fundamentally) revolutionized what is our understanding of physics/physical experience.
Compare the setting and orange Sun with the fully illuminated and setting/WHITE Moon DIRECTLY. They do basically appear to give off the same illumination, in fact. Notice that the curvature or shape of said Moon matches that of THE EARTH/ground (given a clear horizon, that is). What is THE SUN AND what is THE MOON are then the SAME SIZE in the sky as what is THE EYE. Moreover, these two forms manifest (or form up) at what is EYE LEVEL/body height. GREAT !!! Now, I have CLEARLY proven why AND how this is so. The tides are (CLEARLY AND NECESSARILY) ELECTROMAGNETIC/gravitational ON BALANCE. Magnificent !!! The tides are CLEARLY and necessarily subject to F=ma AND E=mc2, as ELECTROMAGNETISM/energy is gravity (ON/IN BALANCE). Again, the stars AND PLANETS are POINTS in the night sky; and the rotation of WHAT IS THE MOON matches it's revolution. The sky is blue, and THE EARTH is ALSO BLUE. Excellent. I have explained why the sizes of said Moon and said Sun are the same as what is THE EYE ON BALANCE. So, I have CLEARLY explained why the diameter of the Moon is about one quarter (at 27 percent) in size compared with what is THE EARTH.
How does what is the Sun survive or exist against what is outer “space" ? The answer is, ON BALANCE, invisible AND VISIBLE SPACE in fundamental equilibrium and BALANCE. Here's why. (Think about TIME as well.)
WHAT IS THE EXPLANATION OF SPACE AND TIME ON BALANCE:
Invisible AND VISIBLE SPACE in fundamental equilibrium AND BALANCE. This necessarily and clearly involves interaction, on balance. Consider what is the eye (on balance). Logically consider what is a two dimensional surface OR SPACE ON BALANCE !!!! Notice the associated DOME AND the flat/black “space” of WHAT IS the eye AS WELL. Really think about it all. Consider WHAT IS THE SUN ON BALANCE. (NOW, think about time.) Outer “space” involves full inertia, AND it is fully invisible AND black. Consider one and three dimensions ON BALANCE !!! Now, consider what is the fourth dimension and the term c4 from Einstein's field equations. Think about ELECTROMAGNETISM/energy AND think about gravity (ON/IN BALANCE). Consider what is the man (AND THE EYE ON BALANCE) who IS actually standing ON what is THE EARTH/ground (ON BALANCE) !! Think about TIME !! Think about why there is something instead of nothing ON BALANCE. Consider that time is NECESSARILY possible/potential AND actual ON/IN BALANCE. Think about the man (THE EYE) that actually IS IN what is outer “space”. Think about time. The stars AND PLANETS are POINTS in the night sky ON BALANCE, and consider what is the speed of light (c) ON BALANCE. NOW, consider what is THE SUN. Think closely about everything in this writing. Balance and completeness go hand in hand. Magnificent. ❤️
c squared CLEARLY means an INTERACTION on balance, as the stars AND PLANETS are POINTS in the night sky. Balanced inertia/INERTIAL RESISTANCE is fundamental. Consider E and “m" on balance. Great. The fourth dimension is only consistent with what is (on balance) a TWO dimensional surface OR SPACE ON BALANCE. Consider what is the eye. Consider what is the balanced MIDDLE DISTANCE in/of SPACE (ON/IN BALANCE). Consider invisible AND VISIBLE SPACE in fundamental equilibrium AND BALANCE. The stars AND PLANETS are POINTS in the night sky ON BALANCE. So, consider what is the speed of light (c) ON BALANCE. This CLEARLY explains the fourth dimension AND the term c4 from Einstein's field equations.
Time is NECESSARILY possible/potential AND actual ON/IN BALANCE, as BALANCED inertia/INERTIAL RESISTANCE is fundamental pursuant to F=ma AND E=mc2 in balance. A galaxy consists of invisible AND VISIBLE SPACE in fundamental equilibrium and BALANCE, thereby eliminating the need for any "dark" "matter" or "dark" "energy"; as ELECTROMAGNETISM/energy is then CLEARLY gravity ON/in BALANCE.
By Frank DiMeglio
DiMeglio means real physics.
Strang killing it again.
What a nice explanation, thank you so much professor :)
30-01-2023.A good exaple of a deformation of a object in time, were the eleaments are noted in a 3-D Matrix, just saying.
Thanks professor
Many thanks
Thank you very much! It was helpful!!
Good job. Right to the point.
Thank you....
Laplace transform is one of the best things calculus has in itself
gay bullahit it is
Great Lecture!👌
Helpful, thanks.
he is a rockstar
This is an amazing lecture, thanks 🙏 so much sir
Is there a link to the video he references talking about the input and growth function over the remaining time?
thanks! this is very helpful :)
Ok genius iam also approaching the problem same way like you I don't use matheMatical way my question is so simple because LTI depends on convulution here's my question below
Convulution is nothing but stacks and scales the input that's why the input to an amplifier is stacked and scaled or amplified but in filter design it attenuate frequency so I don't know how it regret certain frequency by stacking and scaling the input if possible some one explain to me
Also application
Brother I know mechanical engineers could find resonance but when I had a deep thought on this resonance Is an slow accumulation of energy which is accumulated very high in small installments when the frequencies match if you strike a turning fork of 50 hz you get the same frequency of vibration on another tuning fork so they both vibrate if you strike it harder the amplitude changes hence loudness is a human factor the frequency is the same the languages that human speak through out the world the sound only resonate your ear drum for few seconds my question is that the harmonics is the fundamental frequency and overtones are the frequency that follow it take a word in any language you spell it according to convolution the thing scales and ques and stack the signal so convolution can be used to model resonance so when your ear drum vibrates it vibrates so the electrical signals are carried to brain like tuning fork ear drums vibrate within the audible spectrum 20 hz to 20000 hz hence resonance is caused by the word we speak and within the audible range the ear drums vibrate and we make sense of words I have seen in one videos on TH-cam that due to harmonics in any sound causes resonance which could be modelled by convolution recalling the resonance its destructive because slow and steady accumulation of sound on the mass causes high stress and high energy to build inside and stress increase and the system fractures or collapses but our ear drum hearing the sound from human languages try to vibrate but why our ear drum when subjected to continuous exposure of sound does not fracture or rupture like a wine glass iam not telling about high loud sound higher than 80 db but a audible range sound within the frequency of 20 hz to 20000 hz under continuous exposure why it's not damaging it again not failure by high energy but low one in synchronisation on air . But I tried it in my students when I told them to be quite in class they did not listen to me so I took my phone and set an frequency 14000 hz and they told it was irritating the idea of resonance is "small effort but large destruction " just like Tacoma bridge where the wind just slowly accumulated energy on the bridge and it collapsed it so my conclusion is if an audible frequency at continuous exposure to an human ear can it cause bleeding again "small effort but great destruction" sorry for the long story I you are able to reach hear you must be as curious as me so still not finished the ear drum is shook by harmonics in the sound we make by the words( or )overtones in the sound we make by the words I know harmonics is the fundamental frequency and overtones are following it which under slow and steady accumulation of sound energy resonates and could damge the ear drums again "small effort but big destruction" not to mention we assume the person is in coma or brain dead hence when the sound irritates him he or she could not make a move so my question is so simple normally human ear responds to harmonics or overtones according to convolution which could be a disaster but with minimal effort 🙏🙏🙏🙏 at here I could be wrong because harmonics can also be used to construct sound so can it be destructive or the overtones which are the trouble makers and which one according to this gives a response curve when two signals convolved by harmonics or overtones which is destructive but with minimal effort and convolution happens when ear drums oscillate is by harmonics or the overtones or also the trouble makers there
has no application to this situation whatsoever, destructive and constructive effects don't apply at all in real world scales, go find out what the meaning of harmonics and overtone is and how u define pain is absurd too, implying pain to damage is bullshit
wowow, great question!!!
fenomenal
Amazing
Brilliant !
i wish herbert gross had one on transforms
He has
now I know where the operator convolution comes from.
6:38 is wrong isn't it? Transform of y' should be sY(s) - y(0)
tripp he said he isn’t taking the boundary conditions.
Well yes, but maybe he's assuming that y(0) is just simply 0 only to make the example easier
0:30
Samuel Coral Nuño if it were zero it wont be there. Theres clearly a mistake in his statement. It happens to the best of us.
Zero boundary conditions, that's the same reason the first 2nd order ODE yields (s^2+sB+C)Y(s)=F(s)
What book did he reference?
This video series develops those subjects both separately and together and supplements Prof. Strang's textbook, Differential Equations and Linear Algebra: www-math.mit.edu/~gs/dela/
It's Dr. Frankenstein!
I kid, I kid--he's cool.
thank u
Nycc sir
ahhh fuck...the word convolution triggered my PTSD.
ok
Why am I wasting thousands of dollars in uni when I can learn all this for free.....
Khan Academy is good too
What book do we use?
It's Prof. Strang's book: Differential Equations and Linear Algebra www-math.mit.edu/~gs/dela/, you can check out the complete OCW course site for more information: ocw.mit.edu/RES-18-009F15.
Oh, Thank You. I see Complete Manual and [Page of Solution Formulas] those are downloadable for free.
@@mitocw you're excellent
So, calculus (integration) is easier than algebra (partial fractions) xD
Often times, the point is to use the Laplace transform, to avoid the need to do the convolution integral. A typical example where it may seem that the Laplace transform cannot work with partial fractions, is convolving combinations of either sine with itself, cosine with itself, or the two with each other. Partial fractions directly do not help you, but there is an alternative method to still work in the Laplace domain and solve it.
Consider sin(t) conv sin(t). This sets up the Laplace transform of 1/(s^2 + 1)^2. You could do it by convolution integral using many trig identities, or you could do it by anticipating the form of the solution in the time domain, and matching coefficients to the corresponding Laplace domain.
Assume the solution is a linear combination of sin(t), cos(t), t*sin(t), and t*cos(t):
A*sin(t) + B*cos(t) + C*t*sin(t) + D*t*cos(t)
We know £{sin(t)} and £{cos(t)} from a reference table, but need the s-derivative property to find it for the remaining two functions:
£{t*f(t)} = -d/ds £{f(t)}
£{t*sin(t)} = -d/ds 1/(s^2 + 1) = 2*s/(s^2 + 1)^2
£{t*cos(t)} = -d/ds s/(s^2 + 1) = (s^2 - 1)/(s^2 + 1)^2
Thus:
1/(s^2 + 1)^2 = A/(s^2 + 1) + B*s/(s^2 + 1) + C*2*s/(s^2 + 1)^2 + D*(s^2 - 1)/(s^2 + 1)^2
Clear fractions, expand, and gather:
1 = A*(s^2 + 1) + B*s*(s^2 + 1) + C*2*s + D*(s^2 - 1)
1 = A*s^2 + A + B*s^3 + B*s + C*2*s + D*s^2 - D
1 = (A + D)*s^2 + B*s^3 + (B + 2*C)*s + (A - D)
Equate corresponding coefs:
A + D = 0; B = 0; B + 2*C = 0; A - D = 1
Solution for A, B, C, & D:
A = 1/2, B=0, C=0, D = -1/2
Thus, the solution is:
1/2*sin(t) + 1/2*t*cos(t)
He got a negative wrong at the end. The e^at should be first.
He got it right