Text solve y 50y 625y 0 y0 4 y0 9 y0 90 y0 175 70345

Name: text solve y 50y 625y 0 y0 4 y0 9 y0 90 y0 175 70345
Uploaded: 2022-03-03T23:58:11-08:00
Duration: 12 min 25 s
Channel: Ekaveera Kumar
Description: text solve y 50y 625y 0 y0 4 y0 9 y0 90 y0 175 70345

Transcript

00:01 This is the fourth order differential equation.

00:03 Let's use laplace transforms.

00:05 So using the formula laplace transform of nth derivative is s power nf of s minus s power n minus 1 f of 0 minus s power n minus 1 f of 0 minus s power n minus 1 minus 2 f dash 0 evaluated at 0 minus 0 minus f and minus 1 the derivative at 0.

00:26 So laplace transform of 4th derivative at 0 will be s power 4 y of s minus s cube y at 0 minus s square y dash 0 minus s y double dash 0 minus s y double dash 0 minus y triple dash plugging in the values so you'll get s power 4 y of s minus s cube into y of zero what is y of 0 is y of 0 given negative 4 so plus 4 s cube and plus 6x square minus 70 s minus 200 next what is laplace transform of second derivative it is s square y of s minus s y of 0 minus s it's 0 so it is s square y of s minus y of 0 is negative so plus 4 s plus 6 so technically laplace transform with the differential equation d power 4 y by d x power 4 plus 50 d square y by d x square plus 625 y is equal to 0 so we will be getting s power 4 y of s plus 4 s cube plus 6 s square minus 70s minus 200 plus 50 times s square y of s plus 4 s plus 6 plus 625 y of s is equal to 0 so collecting all y of s 1 place so y of s into s power 4 plus 50 s square plus 625 and the other terms are 4 s cube plus 6s minus 70 s minus 200 plus 200 s plus 300 plus 300 plus 300 is equal to 0 so y of s into s power 4 plus 50 s square plus 625 is equal to negative of 4 s cube and 6 s square what is the 200 s minus 70s 130 s right so it is minus one 30s plus one that side we take and 300 minus 200 is a hundred so plus hundred so finally y of s is minus of 4 s cube plus 6 s square plus 130 s plus hundred all divided by this one this i can write it as s square plus 25 the whole square so now we need to split this into partial fractions so the numerator i can write it as minus of can i write the numerator or something like this 4 s into s square plus second sorry yeah 4 s into s square plus 25 that means 4 s q plus 100 s means extra i need 30s so 6 s square plus 30s divided by this is square plus 25 whole square so let's do something else negative of 4 s into s square plus 25 can i write 6x square plus 30 s plus 100 as 6 bracket s square plus 25 plus 30s minus 50 whole divided by s square plus 25 whole square so basically the top cubic one i have have written in this way 4 sq plus 130s plus 6x square plus hundred splitting so it is negative 4 s divided by s square plus 25 negative 6 by s square plus 25 and finally negative of 30 s minus 50 divided by this is square plus 25 whole square so this is mine laplace transform i mean y of us taking the inverse laplace transform y of t will be negative 4 into inverse laplace transform of s by s square plus 25 minus 6 into inverse laplace transform of i can write these are 6 by 5 to 5 by s square plus 25 minus inverse laplace transform of 30 s minus 50 divided by s square plus 25 the whole square what is inverse laplace transform of s by s square plus 25 is minus 4 cost 5 t minus 6 by 5 times sine 5 t minus some g of t so let's quickly find the inverse laplace transform of 30s minus 50 divided by s square plus 25 whole square so how to find the inverse laplace transform of this now remember there are two nice formulas laplace transform of t cost a t u of t is s square minus a square divided by square plus a square whole square and laplace transform of t sine a t u of t will be basically it will be the derivative of negative derivative of a by s square plus a square so what is the derivative of this is a negative a negative 1 by s square plus a square whole square into 2 s so basically it will be 2a s by s square plus a square whole square this is the laplace transform of t sine a t u of t now laplace transform of t cost a t u of t actually i can write it as s square plus a square minus two a square divided by s square plus a square whole square which is actually one by s square plus a square minus two a square divided by s square plus a square whole square sorry so that means laplace transform of t cos a t u of t will be what is this this is actually one by a laplace transform of uh sine a t minus 2 a square divided by s square plus a square whole square so that implies laplace transform of t cos a t for u of t yeah t cause a t 1 by a sign a t u of t is negative 2a squared divided by c square plus a square whole square and also this 1 by 2a if you take it below a plus a transform of 1 by 2a whole square.

08:58 Now we need to find the inverse laplace transform of 30 s minus 50 divided by s square plus 25 whole square which i can add it as 30 times inverse laplace transformer s by s square plus 25 whole square plus inverse laplace transform of 50 i can add it as negative 2 into 25 divided by s square plus 25 whole square so using the above formula, i'll be getting 30 times 1 by 2a...

Question

Text: Solve y'''' + 50y'' + 625y = 0 y(0) = -4, y'(0) = -9, y''(0) = 90, y'''(0) = 175

Question

Text: Solve y'''' + 50y'' + 625y = 0 y(0) = -4, y'(0) = -9, y''(0) = 90, y'''(0) = 175

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