00:01
Let's solve for the following laplace transforms.
00:04
A laplace transform of cosine of 3t plus 5e to the negative t.
00:10
We know that the laplace operator could be distributed on summation.
00:15
So let's write this as laplace transform of cosine of 3t plus 5 laplace transform of e to the negative t.
00:25
Note that the 5 times laplace transform of e to the negative t is equal to laplace transform of 5 times e to the negative t.
00:35
Which could be taken out due to the linearity property of the laplace transform.
00:42
So here let's recall some laplace formulas.
00:49
Laplace transform of cosine of omega t is equal to s over s squared plus omega squared.
00:55
And laplace transform of e to the at is equal to 1 over s minus a.
01:06
So this is equal to s over s squared plus 9 plus 5 over s plus 1.
01:19
B laplace transform of e to the negative 3t times 2t squared minus 5t plus 1.
01:28
Let's rewrite this laplace transform as 2 laplace transform of e to the negative 3t times t squared minus 5 laplace transform of e to the negative 3t times t plus laplace transform of e to the negative 3t.
01:47
Now let's recall that if laplace transform of f of t is equal to capital f of s.
02:01
Then the laplace transform of e to the at times f of t is equal to capital f of s minus a.
02:25
We know that the laplace transform of t squared is 1 over s cubed.
02:37
So the laplace transform of e to the negative 3t t squared is equal to 1 over s plus 3 cubed.
02:48
The laplace transform of t is 1 over s squared.
02:54
So the laplace transform of e to the negative 3t times t is equal to 1 over s plus 3 squared.
03:04
And the laplace transform of e to the negative 3t is 1 over s plus 3.
03:10
So this expression is equal to 2 over s plus 3 cubed minus 5 over s plus 3 squared plus 1 over s plus 3.
03:32
C the laplace transform of sine of 70 minus hyperbolic cosine of 3t.
03:46
Which just like a and b could be written as laplace transform of sine of 70 minus laplace transform of hyperbolic cosine of 3t.
04:01
Note that the laplace transform of sine of at is equal to a over s squared plus a squared.
04:11
So laplace transform of sine of 70 is equal to 7 over s squared plus 49...
