Reduction of orders, 2nd order differential equations with variable coefficients
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- Опубликовано: 27 авг 2024
- Differential equation tutorial on 2nd order differential equations with variable coefficients. We will solve x^2y''+3xy'-8y=0 by using the reduction of orders method. This is actually an example of the Cauchy-Euler differential equation. See • Cauchy Euler Different... for the introduction.
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I have my first ode test Wednesday. Been doing math for 12 hours. You and the organic chemist are the biggest help in the world.
Cannot get over how seamlessly he switches pens. Asian level: over 9000
this is literally the only video on youtube showcasing second order DEs with variable coefficients. bravo. you're not variablephobic, liked.
this guy is amazing. my prof left but forgot to put up vid on reduction of order and i just knew blackpenredpen would be here to save the day and help me through the vague lecture notes
These videos really help me beyond belief. Thank you so much!
you are an absolute genius. I have my Calc test in a couple of days and your amazing videos help a lot! Keep up this great vids, love them!
you do the best job when it comes to solving difficult problems, your examples are not easy as 1+1. I feel better prepared after seeing you solving this, now i will try myself. Thanks!
Glad to hear!!! Thanks!!!
@@blackpenredpen Wait a minute..where did you get that expresdion at thenbeginning from..the equation for y2..it looks nothing like the original euation with t squared and t and y..please explain
You're really helping me understand and successfully pass differential equations in my college, thank you very much!
At first i thought: "how hard could it be, its math after all."
now i am striving to pass
that intro :) WHY AM I SO INTERESTED IN 2ND ORDER DIFFERENTIAL EQUATIONS :D
One of the best online maths tutor I wish u was my maths lecturer
Thanks!
I am a teacher actually, but let's keep that a secret. : )
you are single-handedly saving my gpa. thank you
thanx for your videos. In your opinion how to make kids not only interested in Math but also creative in dealing with problems.
Thank you for this! Clarity makes your video easy to understand.
I didn't understand this topic in class, but after this video I think I can forever use the Reduction of order method to find the second solution without any doubts. Thank you so much, you also made this video exciting, short, and easy to understand.
for those who are confused...ln|t^-7|+k1 ==> ln|t^-7|+lnk2 (so we have k1==lnk2, k2>0)...and then we use that logarithm rule: ln(a)+ln(b)==ln(ab)..so we get ln(k2*|t^-7|)
Thanks man. This helped me a lot. What I would really like to see, Is one of these equations solved in various ways. So we can compare which way would be the best to use
This is fantastic! If you let y1 be a solution to a second order homogeneous equation with variable coefficients in standard form, and work your way through this process it leads to a real neat formula you can use
You''re a good man, really very good!! I appreciate what you've done for us
at 12:48 the new guy came along with addition so he easily did that but what if it comes as multiplication in denominator and what if there's an e too in multiplication with a long power
genius. I like your explanation. awesome
Love your videos. Even older ones like this one
Also you're the first person in a while that I've seen write seven as " 7 " and not " -7- "
Could you explain in more detail why the absolute value bars do not matter?
Thank you so much. It helps me a lot.Before this video, i have no idea about reduction of order but now i am pretty sure i know it.
hardest carry ever my prof couldn't explain shit about reduction of order but this video was so clear !
Thanks buddy, love from nepal
Also y=0 is a solution although it’s trivial you don’t obtain it from the general solution because even if we let c1and c2 equal 0, we still have a case when t=0 is undefined in the general solution but y=0 is for all values of t including t=0
great presentation dawg, you a real one
You do it so much better than my teacher does it.
Autumn Knight thank you!
you are a real hero!!!!!
but y and y2 and different things
how can we replace y by y2
Very nicely explained
What happened to your constant 1/6 in your last integral?
It got multiplied by the other constant k2 and became yet another unknown constant k3
In other words 1/6(k2)=k3 (simply another constant we dont know)
It helped to my homework, thank you so much!
I owe you my college tuition
Thank you BlackpenRedPenBluePen
I got test in 2 hours and this is a lot to absorb.
best of luck
@@blackpenredpen Well as a part time student in NTU, i think i did fairly well consider the short amount of time. I learned about integration, first order of derivatives from you and it turned out well xD
However, I still couldn't figure out them series chapter, too difficult and like I say, I'm a part time student who does job at an oil and gas industry during the day time.
Thanks blackpenredpen for your assistance :)
This whole playlist is great mate ha ha. Again thanks
What if t^2 is not a given? Can you still solve this ?
yeah, do some y=x^r substitution
it is in fact a cauchy-euler differential equation
@@franciscovillicana2130 is this what yields C1e^(r1x) + C2e^(r2x)?
Yes it's solved by Cauchy differential equations so it has solved by characteristics...
Love these videos so much
I prefer the Cauchy Euler strat, is there a case where this is a more optimal approach?
Differential equations can be solved with many ways, truly this equation is a C.E case, but there’s another option given by this video, it’s helpful for anyone who’s learning Differential equation.
@@Smallsh123 alright epic thank you 😁 Cauchy Euler op 😂
we can also rely on abel’s theorem to solve this eqn. Would reduce some algebra, but i’m not sure if that would be defined as the “reduction of order’’ technique :)
Very easy to understand.
would be easier to let y = X^r then take derivative and second derivative and plug in .... you factor out X^r and then you solve for zero the same you would an easy 2nd order homogenous equation with constant variables
why he take y=t^2 please
Thank you 😊
Y1 is given to be t^2, ok. But how does it imply that Y2 be like V(t) * t^2 ?
Does this method still work if you don't know that t^2 is a solution at the beginning?
What if you don't know any solutions to the DE in advance?
what if i don't have any solutions like t^2 at all in the beginning?
Nicely done.
Our first reduction of order problem has us reduce a nonlinear second-order ODE where the solution function u(t) is vector-valued with 2 components. I am a bit lost xD
I get how you use a solution to solve the non autonomous equation, but how do you find the first solution? I tried laplace transform, variation of parameters, power series, diagonalization. I still can’t seem to solve the general form y’’+Py’+Q
Thank you thank you thank you thank you
Thank you so much this was very helpful !!
Thanks pal
Thank You .. It's all clear now ..!
the logic is pretty clear
good explanation! thank you.
thank you this was very helpful.
GREAT VIDEO!!! thank you!
Thank u so much sir, keep going ❤
Thank you , thank you , thank you , thank you , thank you ........♥️
Nice video its very helpful
can i ask before using reduction order, there is cauchy-euler equation/ for example (x-1)y"-xy'+y=0 here how to make it possible to use reduction order?
Academic Videos hey thanks for replying my question, can you explain how to guess the first solution more detail? I just realize that reduction order is for finding second solution
Why don't the absolute values matter? Do they always not matter for problems like this?
i have a black pen, i have a red pen, ughh, blackpenredpen
Why is x^2 and x in original equation changed to t^2 and t?
Thank you sir
why we named reduction of order please answer me
because we are using the substitution of w= v' and w'= v''. This reduces it from a second order ODE to a first order, which can easily be solved using separation of variables or integrating factor. RoO can be used for higher orders as well, though more substitutions would be needed
thank you so much for your vedeos
Where did you get Y2
what if there was a t^2 coefficient with y term also?
Can also solve this by cauchy euler, must faster.
Help! How do you solve the same equation if you aren't given t^2 as a solution in the first place?
You can check my video on “Cauchy-Euler equation”
Can anyone explaine
?why y=v(t)t^2
So cool
I kinda forgot the deal 😿 why the absolute value doesn't matter?
To make them linearly indepent.
how u will solve the equation when t^2 is not given ..
Do you know how to make this exercise: x^3y```-x^2y``+2xy`-2y=x^2 , y=y(x) Thank you :))
non homogeneous 3rd order ODE.y=yh+yp
I use inverse differental operator (D+p)y=q -> [(e^px)*y] ' = q*(e^px) -> (e^(px) *y)=Integral (q*(e^px) dx)
y=[e^(-px)] [Integral (q*(e^px) dx)]
Can be written y=(D+p)^(-1)[q]
nth order ODE,downgrade to 1st order linear ODE.Use exact 1st order ODE to solve .
x=e^t t=ln(x) (x^2)=e^(2t)
d/dx=(d/dt) (dt/dx)= 1/x (d/dt) ->xy'=dy/dt
d^2/d(x^2)=(d/dx) (d/dx)=(d/dx) ((1/x) (d/dt))=(-1/t^2) (d/dt)+(1/t^2) (d^2/d t^2)->(x^2) y"= ((d^2)/d (t^2)) y - (d/dt) y
d^3/(d x^3)=(d/dx) (d^2/d x^2)= (1/x^3) ((d^3)/d (t^3)- 3/(x^3) (d^2/dt^2)+(2/x^3) (d/dt)
->(x^3)y"'=((d^3)/d(t^3))y-3((d^2)/d(t^2))y+2(d/dt)y ----> d/dt=D(important)
(x^3)y'"-(x^2)y"+2xy'-2y=x^2 ----> {[D^3-3(D^2)+2D]-(D^2-D)+(2*2D)-2}y=e^(2t)
(D^3-4(D^2)+5D-2)y=e^(2t)
[(D-1) (D-1) (D-2)]y=e^(2t)
yh(homogeneous)=c1 (e^t) +c2 (t) (e^t)+c3 [e^(2t)]
start to solve yp
(D-1) (D-1) (D-2) y = e^2t , (D-1) (D-2) y=y1 (3rd downgrade to 1st)
(D-1)y1=e^2t
[(e^(-t)*y1] ' = (e^2t)*((e^(-t))=e^t
(e^(-t) *y1)=Integral (e^t) dt=e^t
y1=e^2t
(D-1)(D-2) y=y1 ->( D-1)(D-2)y=e^2t
(D-2)y=y2 (2nd downgrade to 1st)
(D-1)y2=e^2t
[(e^(-t)*y2] ' = (e^2t)*((e^(-t))=e^t
(e^(-t) *y2)=Integral (e^t) dt=e^t
y2=e^2t
(D-2)y=y2 (1st order)
(D-2)y=e^2t
[(e^(-2t)*y] ' = (e^2t)*((e^(-2t))=1
(e^(-2t) *y)=Integral (1) dt=t
y=t*(e^2t)
so yp=t*(e^2t)
y=yh+yp=c1*(e^t)+c2* t *( e^t)+c3 * (e^2t) +t *(e ^2t)
In college I disliked differential equations. Now I watch them for fun.
I knew you would substitute k_3!
Wonderful...
What if a solution is not given initially?
Tornado of Terror
You can use the “Cauchy Euler” approach. I have a video on that. You can check it out.
i really need your help i have an integral i coudnt solve it can you help me please
7:50 I died laughing
hahaha
Do you want nice equation to solve ?
x^4 d^2y/dx^2+(4x^3+x^2) dy/dx +(2x^2+x-1)y=-x
There could be problems with integrals
but it is solvable in standard way
If the right hand side is not equal to zero..If it is a function of x..how can it be solved?
I love you
And i didn't know about k_4!
I want you to safe me from a mental breakdown 🙁please reply and help meer sir
just for fun
thug life
DAAABLUE
Hahahha thanks master! ♥♥♥♥
Why is he holding a Pokemon ball?
wondering the same, this curiosity is killing me
This is the microphone
"dabalu" ang kyot
The video is great but the fuck it feels like i am watching an anime xD
I think u forgot the constant - 1/6
He multiplied it to k2 (which is unknown constant) and became k3 which is STILL unknown constant
very amazly hot
These differenciale equations are more boring! :( :( :( ...
shush
Ray Lai u got my heart too