Solve the differential equations using the laplace transform...

Transcript

00:01 Hello students in this question we need to solve the given differential equation by using the laplace transform given that d squared y by d t squared plus 4 d y by d t plus 3y equals e power minus 2 t initial conditions are y of 0 equals 1 and y dash of 0 equals 0 on applying the laplas transform then we get yes square y bar minus s y of 0 minus y dash of 0 plus 4 s y bar minus 4 y of 0 plus 3 y bar which is equal to e power minus 2t can be written as 1 by s plus 2 by using the laplace transform it can be written as s square y bar minus s plus 4 s y bar minus 4 plus 3 y bar equals 1 by s plus 2.

01:10 It can be written as y bar equals 1 by s plus 2 into s square plus 4 s plus 3 plus s divided by s 2 plus 4 s plus 3 plus 0 plus 4 divider by s 2 plus 2 into s 2 plus 1 into s plus 3.

01:19 On facturizing we get 1 by s plus 2 into s plus 2 into s plus 3 plus s 2 into s plus 3.

01:29 On applying the partial fraction then we get y bar equal equal 2 2 into s plus 3 plus 1 into s plus 3.

01:32 Minus 1 by s plus 2 plus 1 by 2 into s plus 1 plus 1 by 2 into s plus 3 plus 3 minus 1 by 2 into s plus 1 plus 1 plus 2 by s plus 1.

01:48 On applying the laplice transform then we get l inverse of y bar which is equal to e power minus 2 t minus plus 1 by 2 e power minus 2 t minus plus 1 by 2 e power minus t plus 1 by 2 e power minus 3 t plus 3 by 2 e power minus 3 t minus 1 1 by 2 e power minus t plus 4 into 2 e power minus t on further simplification we get the answer as y of t equals minus e power minus 2 t plus 10 e power minus 3 t plus 8 e power minus t for b to question given that d squared y by d t squared plus d y by d t plus 3 y equals sine 2 t an initial condition y of 0 equals 1 and y dash of 0 on applying the laplace transform we get s squared y bar minus s y of 0 minus y dash of 0 plus 4 into s y bar minus 4 into y of 0 plus 3 y bar which is equal to on applying the laplace transform we get 2 divided by s square plus 2 square that is 4 therefore on applying the initial condition then we can simplify as s square plus 4 s plus 3 into y bar equals 2 divided by s square plus 4 plus 4 plus s plus s plus 4 plus s plus 4 this term can be simplified as y bar equals 2 divided by s squared plus 4 plus 4 s plus 3 plus s divided by s squared plus 4 s plus 3 plus 4 divided by s squared plus 4 s plus 3...

Question

Solve the differential equations using the Laplace transform: a) d²y/dt² + 4dy/dt + 3y = e^(-2t), y(0) = 1, y'(0) = 0 b) d²y/dt² + 4dy/dt + 3y = sin(2t), y(0) = 1, y'(0) = 0

Question

Solve the differential equations using the Laplace transform: a) d²y/dt² + 4dy/dt + 3y = e^(-2t), y(0) = 1, y'(0) = 0 b) d²y/dt² + 4dy/dt + 3y = sin(2t), y(0) = 1, y'(0) = 0

Please give Ace some feedback