Abel's brilliant trick for solving differential equations

Abel's method is an interesting approach to solving 2nd order linear differential equations using the Wronskian. Here's a nice example of its application.

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[orionspur]

Totally guessed sin(x)arccoth(sin(x))-1 out of thin air while both cooking and mowing my lawn.

[mcalkis]

What a brilliant method. Thank you for showing this

[CM_France]

Hi,

"ok, cool" : 3:42, 7:12, 7:40, 9:18 .

"terribly sorry about that" : 5:24 .

[aliaujla]

Bro please make a video in detail about cauchy residue theorem.

[randomzhjioewmx]

Here is another (essentially the same) method to obtain y_2 knowing already y_1=sin(x).
We aim to factor the operator D^2-tan(x)D+2. This operator vanishes at sin(x), which also is a zero of the operator D-cot(x), so one may use the ansatz
Multiplying out and using Dcot(x)=cot(x)D-1/sin^2(x), one obtains Comparing coefficients, tan(x)=f+cot(x) so f=tan(x)-cot(x). [One may check that this satisfies 2=fcot(x)+1/sin^2(x) so the factorization works].
So to find another zero h(x) of the operator (D-f)(D-cot(x)), we can first find a zero g(x) of D-f, and then solve (D-cot(x))h=g.
To find g, one must solve g'-fg=0, so g'/g=f, so [we wlog choose 0 for the constant of integration) so g=1/(sin(x)cos(x)).
To find h, we must solve the linear equation h'-cot(x)h=g. We make ansatz h(x)=sin(x)m(x). Thus sin(x)m'(x)=g, so m'(x)=1/(sin^2(x)cos(x)). Integration yields Thus sin(x).
So one may set y_2=sin(x)arctanh(sin(x))-1.

[bandishrupnath]

Sir pls make some videos which discusses different types of things like melin transform, laplace transformation as some of ur viewers are illiterate(Me also especially) compared to ur knowledge in calculus 🙃

[AdrianRif]

Very elegant solution, but I think to make the equation more general you should have multiplied the hyperbolic cotangent term inside the parenthesis by the constant B as well!

[pandavroomvroom]

"your fav integrator on youtube" - well its true

[gideonbrown]

Wow reducing the second order to the first order using any arbitrary w where w is the wronskian is super cool…😍🥰🥰🥰😍😍

[dharunpranay]

I am feeling ill now for studying engineering .But I love maths the most than anything. But our India is not conducting such competitions like mit this makes me 😢

[fartoxedm]

I can't sleep well when B's at the end disappear... Absolutely fascinating technique tho! Thanks for sharing

[pokakoka]

Brilliant sir, sir just tell me the name of app you are using for video recordings. Will wait sir

[alexander_elektronik]

i think you forgot to multiply the C_2 with y_1 at 9:33 but since you set c_2 = 0 it wouldn‘t make a difference :)

[usmansaleem]

Kindly mention white board software you are using is also the name of the pen tablet

[alecbg]

This channel is awesome. 10:20 should be C_2 sin(x) right?

[gerryiles]

In your final answer, the coth-1 should have been multiplied by B...

[farfa]

Since x only shows as sinx, would it be correct to ditch the sin and change the domain to [-1, 1]?

[jieyuenlee]

9:56 Int(secx/sin²x)dx
=Int(secxcsc²x)dx

=-1/(tanxcosx)+ln(secx+tanx)
=-1/sinx+ln(secx+tanx)
=ln(secx+tanx)-cscx
=ln((1+sinx)/cosx)-cscx
=1/2ln[(1+sinx)²/cos²x]-cscx

[maddog]

Am I missing something? The arccoth(x) is not real for abs(x)< 1, which is pretty much where the sin(x) lives. If you want to include complex solutions, fine. But I have a feeling we were dealing with real quantities.

[sgiri]

Sir, can this differential equation gives answer when we try the series solution or frobenius solution depending upon the case?