6.10 Using the transform pairs in Table 6.1 and the...

6.10 Using the transform pairs in Table 6.1 and the properties of the Laplace transform, prove the following Laplace transform pairs: (a) t cos(??t)u(t) ? (s² - ??²) / (s² + a²)²; (b) t sin(??t)u(t) ? 2??s / (s² + a²)²; (c) 1/(2a³) (sin(at) - at cos(at))u(t) ? 1 / (s² + a²)².

Transcript

00:01 Hello students in this question.

00:02 We have to prove the laplace transform of given functions.

00:06 Here are three functions given.

00:08 The first is t cos omega naught t 2 of t laplace transform s square minus omega naught t omega naught square upon s square plus a square whole square but on the right hand side in place of a is given omega naught.

00:38 So we will write here s square minus omega naught square upon s square plus omega naught square whole square.

00:51 Now on the left hand side, we will take the laplace transform.

00:56 So first of all laplace transform of t cos omega naught t 2 of t is a function.

01:11 So this will be equal to the first of all, we will write the laplace transform of cos omega naught t.

01:19 The laplace transform of cos omega naught t is s upon s square plus omega naught square.

01:32 But when we multiply the t then we write the first differentiation of this function d by ds and minus sign because the formula is minus 1 to the power n tn by tsn of function fs.

01:57 If we write the for like if we multiply t to the power n and of any function like f of t, we take the laplace transform to use this formula.

02:23 Okay.

02:24 Now we will differentiate this function after differentiation.

02:27 We will get and simplification s square minus omega naught square upon s square plus omega naught square the whole square hence proved that the lhs left hand side and the there is a right hand side...

Question

6.10 Using the transform pairs in Table 6.1 and the properties of the Laplace transform, prove the following Laplace transform pairs: (a) t cos(??t)u(t) ? (s² - ??²) / (s² + a²)²; (b) t sin(??t)u(t) ? 2??s / (s² + a²)²; (c) 1/(2a³) (sin(at) - at cos(at))u(t) ? 1 / (s² + a²)².

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