Solving Homogeneous First Order Differential Equations (Differential Equations 21)

Exercises in Solving Homogeneous First Order Differential Equations with Separation of Variables. A review of various integration techniques is included.

[rakeshkottu]

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[IMadeOfClay]

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[MuhammadUmar-dzuw]

I don't know Why I spend my all time in searching for differentials equation..Just got your videos...Ahhh the concept I have taken from you in Calculus 1 and 2, just wanted to say thanks Alot..Stay happy and blessed Sir❤

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[juanmanuelmillansanchez]

This video is amazing, helped me to get the homogenious equations down, i wasnt able to do this on one take though, greetings from colombia, thanks for your amazing content.

[osianoisekenegbe]

1:33:03 You can't _just_ separate the fractions, but you _can_ just separate them anyway 😏, get two fractions, simplify the denominators into v(1+v^2), and use the u-sub u = (1+v^2). This makes the first fraction immediately solvable (but we leave it for later because it can be handled at the same time as another very soon), and with the second, using the concurrent u-sub v^2 = u-1, we get our denominator into a form where we can use linear partial fractions. Then everything cleans up nicely.

(Just an alternative for those who didn't recall how to handle the quadratics in PFD or didn't see that as an option first.)

[Ahwwooooo]

taking this course right now, Leonard always helps! Unfortunately my class is fast paced and we already covered chapters beyond the videos, currently on eigen values and vectors and fundamental solution sets. Looking forward to the next videos!

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[khbye]

Lessons learnt about integration techniques (some of the nitty gritty details, in particular about U sub)

1) Go through a checklist of methods to integrate before jumping into the tough ones (eg trig sub)
[Basic*>separate into fractions which are basic**>U sub>trig sub]***

*can include other techniques other than U-sub like integration by parts, and 'reverse chain rule' (which is really just another special case of U-sub)

**I only know two ways: first way, when it's one big fraction where numerator has multiple terms while denominator has only one. Second way, by partial fractions.

***if you include the techniques learnt so far, then the list continues from trig sub: [separable>linear ODE>homogenous]

2) All about U sub techniques
i)Hate negative signs! Factor them out, and hope you'll see a repeated expression (eg -x^2 -1= -1*(x^2 +1))
ii)Sum of fractions inside a radical? COMBINE those fractions into one (unless...It integrates nicely into the inverse trig or that logarithm one in your table of integrals)
iii)Non-basic integrals with radicals (eg sqroot U) as a factor in the denominator? FORCE that radical to be factorised out (eg sqroot U - U= sqroot U * (1-sqroot U))
iv)Sometimes you need to do U-sub twice! (Or maybe more)
v) When inspired, feel free to change the U substitution and try integrating again.

[isobar]

@44.43...Was always taught this as differential of denominator over denominator...now I understand why...comes from a u sub. Thanks Professor.

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[ericstone]

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