Use Laplace transforms to solve the initial value problems. y"+y'=\delta(t)-\delta(t-3), y(0)=1 , y'(0)=0

Tazmin Horton 2021-09-18 Answered
Use Laplace transforms to solve the initial value problems.
yy=δ(t)δ(t3),y(0)=1,y(0)=0
You can still ask an expert for help

Want to know more about Laplace transform?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

toroztatG
Answered 2021-09-19 Author has 98 answers

L{δ(ta)}=eas
L1{1sa}=eat
L1{eass}=u(ta)
Given
yy=δ(t)δ(t3),y(0)=1,y(0)=0
Taking laplace transformation both sides:
L{y"}+L{y}=L{δ(t)}L{δ(t3)}
[s2Y(s)sy(0)y(0)]+[sy(s)y(0)]=1e3s
(s2y(s)s)+(sy(s)1)=1e3s
y(s)[s2+s]s1=1e3s
y(s)[s2+s]=2e3s+s
y(s)=2+se3ss(s+1)
y(s)=2+ss(s+1)e3ss(s+1)
y(s)=2s1s+1e3ss+e3ss+1
Now taking inverse laplace transformation both sides:
L1{y(s)}=2L1{1s}L1{1s+1}L1{e3ss}+L1{e3ss+1}
y(t)=2×1etu(t3)+e3tu(t3)
y(t)=2et+(e

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-09-04
Find the laplace transform of 12a(sinat+atcosat)
asked 2021-01-10
Sketch the Fourier transforms of the following functions assuming |ω1|>|omega2|
a)f(t)=[cos(ω1t)+isin(ω1t)][cos(ω2t)+isin(ω2t)]
b)f(t)=[cos(ω0t)]etT2
asked 2021-02-24
For the function
f(t)=et
g(t)=e2t
0t<
compute in two different ways:
a) By directly evaluating the integral in the defination of fg
b) By computing L1{F(s)G(s)} where F(s)=L{f(t)} and G(s)=L{g(t)}
asked 2021-03-02

Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by 
(F(s)=0estf(t)dt
where we assume s is a positive real number. For example, to find the Laplace transform off(t)=et , the following improper integral is evaluated using integration by parts:
F(s)=0estetdt=0e(s+1)tdt=1s+1
 Verify the following Laplace transforms, where u is a real number. 
f(t)=tF(s)=1s2

asked 2021-01-13
Discuss is there a solution or not (why) of Laplace transformation of
L(1t)
asked 2020-12-24
Find the laplace transform of the following function
f(t)=tu2(t)
asked 2021-09-19
Find the inverse Laplace transform of the following transfer function:
Y(s)U(s)=50(s+7)2+25
Select one:
a)f(t)=10e7tsin(5t)
b)f(t)=10e7tsin(5t)
c)f(t)=50e7tsin(5t)
d)f(t)=2e7tsin(5t)