Ultimate Inverse Laplace Transform Study Guide

How to do inverse Laplace transform. We will go over 24 inverse Laplace transform with partial fractions and the inverse Laplace transform with completing the square. We will also solve differential equations by Laplace transforms!

Time stamps: 0:00 start
Q1, inverse Laplace transform of 1/s^4, 0:50
Q2, inverse Laplace transform of 1/(6s+3), 3:50
Q3, inverse Laplace transform of (s+1)/(s^2+2), 6:46
Q4, inverse Laplace transform of 1/(s^2+2s) by partial fractions, 11:20
Q5, inverse Laplace transform of s/(s+2)^2, 14:50
Q6, inverse Laplace transform of s*e^(-pi/2*s)/(s^2+1), 25:16
Q7, inverse Laplace transform of s/(s^2+2s+2) with completing the square, 28:45
Q8, inverse Laplace transform of 1/(s^3*(s^2+1)), 36:23
Q9, inverse Laplace transform of 1/(s+2)^5, 48:50
Q10, inverse Laplace transform of 1/sqrt(s)+1/sqrt(e^s), 51:25
Q11, inverse Laplace transform of (s+8)/(s^2+4s+13), 58:03
Q12, inverse Laplace transform of 1/(s^4+5s^2+4), 1:02:58
Q13, inverse Laplace transform of 1/(s^4*e^(10s)), 1:12:00
Q14, inverse Laplace transform of arctan(1/s), 1:15:33
Q15, inverse Laplace transform of ln((s^2+9)/(s^2+1)), 1:25:00
Q16, inverse Laplace transform of 1/(s^4-16), 1:30:56
Q17, inverse Laplace transform of s^3/(s^4-16)^2, 1:43:20
Q18, inverse Laplace transform of 1/(s^4+4s^2+4) by convolution theorem, 1:50:35
Q19, using Laplace transform to solve y'+2y=sin(3t), 2:04:07
Q20, don't watch this one.... 2:14:00
Q21, using Laplace transform to solve a second order diff eq, 2:32:10
Q22, using Laplace transform to solve y''+16y=cos(4t), 2:39:05
Q23, a second-order differential equation with the unit step function, 2:44:11
Q24, yay!!! 2:50:35
LA marathon 2005 medal, 2:55:00

#blackpenredpen #laplace #math

[drpeyam]

At first I thought you were gonna do the Laplace video but in reverse 😂

[frozenmoon]

These marathon videos are becoming my most favourite thing to watch!

[MathswithMuneer]

Hello from a math teacher in Pakistan. I am glad to see teachers taking initiatives and helping students in their problems. I am positive our videos are a great source of help for them. Good work

[vibhupandya]

The hell dude. I just started the original laplace marathon. And ALREADY?

[Giovanni]

These marathons are great, your effort with the worksheet, timestamps, and everything else is greatly appreciated. Helped me out so much.

[blackpenredpen]

All the s's are in red.
How do you distinguish your s and your 5?

[Shailendra]

“Is this heaven?”
- “No this is a Inverse Laplace marathon”
“Hm, fair enough”

[tombartimtim]

It would be nice a double and triple integrals marathon!!

[joshuaokeke]

Someone that understands, S and 5 can be really confusing especially if your handwriting is as bad as mine

[anarbay]

I am taking differential equations in MIT and literally, you are saving my time with excellent exercises. Our book is just awful. Just imagine, some of your exercises appeared in my midterm exam

[lindsaywaterman]

This expression could have been written as 1/8[1/(s^2-4 -2) -1/(s^2+4)] and then 1/16[2/(s^2-4 -2) -2/(s^2+4)]. The Laplace Transform would, therefore be 1/16[Sinh(2t) -Sin(2t)], which is what Black Pen Red Pen got but in a convoluted way.

[thevenin]

Can't I just play the other Laplace video in reverse? :-)

[TheRandomFool]

Coming back and re-watching this video a couple years later, it occurs to me that on question 20 and other hard partial fraction decomposition problems the residue theorem from complex analysis can be used to help with it. You'd just have to calculate a couple derivatives for building up the powers of s, and the rest is fine.

[hevanderdacosta]

Now we just need a fourier and inverse fourier transform marathon.

[humdrumboy]

I have just found ur channel today and hands down ur already one of my favorite teachers on youtube. I wish i knew about u earlier. Ive been studying for some hours now and this is something i didnt do in a very long time. Your videos are very informative and very entertaining.

[OwelleUwaleke]

The beauty of all these videos is that you can watch again, again and again until you come to grasp the concept

[carultch]

Another way to solve the convolution of multiple trig functions:
Based on the degree in the denominator for (s^2 + w^2)^n, the value of (n - 1) tells you how many times you'll eventually multiply trig by t. So you form a linear combination of t^k*sin(w*t) and t^k*cos(w*t), where w is the angular frequency, and k is a power that builds from 0 to (n-1). You then find corresponding Laplace transforms to each of these terms, and add up a linear combination with unknown coefficients, to equate to the original transform.

Use the function parity property of convolution, you can eliminate half of the terms, and have half as many unknowns to solve for.
f_odd(t) conv g_odd(t) = odd function
f_even(t) conv g_even(t) = odd function
f_odd(t) conv g_even(t) = even function

If expecting odd functions, this means you can eliminate all t^even * cos(w*t) terms and t^odd * sin(w*t) terms. Vice versa, if you are expecting even functions. Then you proceed with solving for the unknown coefficients.

[holyshit]

19

Here we can be tricky and build difference of squares from linear factor of denominator
Then we will get constant term if we combine difference of squares with the other factor of denominator
We will get
13=(s^2+9)-(s+2)(s-2)
If we replace numerator by 1/13((s^2+9)-(s+2)(s-2)) we will have nice cancelling


20
16=s^4-(s^2-4)(s^2+4)
and we have nice cancelling
If we know hyperbolic functions we dont need partial fractions

[XgamersXdimensions]

Maybe next could be some linear algebra videos? Ideas could be marathon on: finding Inverses, Eigenvectors, eigenvalues of matrices?

[Amine-gzgq]

I've just finished the Laplace Transform Ultimate Study Guide video now I'm going to start watching this one, it's going to take me a while like the other one because I have other things to do.