Answered question

2022-05-09

 

 

Answer & Explanation

Jazz Frenia

Jazz Frenia

Skilled2023-05-06Added 106 answers

We are given the differential equation:
4d2ydt2+6dydt+5y=t
with initial conditions dydt(0)=y(0)=0.
To find the Laplace transform of y(t), we can apply the Laplace transform to both sides of the differential equation:
4{d2ydt2}+6{dydt}+5{y}={t}
Using the property of the Laplace transform that {d2ydt2}=s2{y}s·y(0)dydt(0) and {dydt}=s{y}y(0), we can rewrite the equation as:
4s2Y(s)4sy(0)4dydt(0)+6sY(s)6y(0)+5Y(s)=1s2
Since dydt(0)=ddt(y(0))=0 and y(0)=0, we can simplify the equation to:
4s2Y(s)+6sY(s)+5Y(s)=1s2
Simplifying further, we get:
(4s2+6s+5)Y(s)=1s2
Therefore, the Laplace transform of y(t) is:
Y(s)=1s2(4s2+6s+5)
To solve for Y(s), we can use partial fraction decomposition. First, we factor the denominator:
4s2+6s+5=(2s+1)2+4
Thus, we can write:
1s2(4s2+6s+5)=As+Bs2+Cs+D2s+1+4Cs+D2s+14
Simplifying, we get:
Y(s)=1s2(As+Bs2+Cs+D2s+1+2Cs+D2s+12)
Y(s)=As+Bs4+Cs+D4s2+4s=As+Bs4+Cs+D4s(s+1)
Using the initial condition y(0)=0, we know that Y(0)=0. Thus, we can solve for B:
Y(0)=B04+D4(0)(0+1)=0B=0
Using the initial condition dydt(0)=0, we know that Y(0)=0. Thus, we can solve for A:
Y(s)=2As34As+Cs2+D4s2(s+1)
Y(0)=00+C(0)+D4(0)(0+1)=0D=0
Now we can write Y(s) in terms of A and C:
Y(s)=Ass4+C4(s+1)
To solve for A and C, we can take the Laplace inverse of Y(s) using tables or the method of partial fraction decomposition:
1{Y(s)}=1{Ass4+C4(s+1)}
y(t)=at3+C4et
where a and C are constants that we still need to solve for.
Using the initial condition dydt(0)=0, we can solve for a:
dydt(t)=3at2C4et
dydt(0)=0=3a(0)2C4e(0)C=0
Thus, the solution to the differential equation with initial conditions is:
y(t)=at3
Using the initial condition y(0)=0, we get:
y(0)=0=a(0)3a=0
Therefore, the solution to the differential equation with initial conditions is y(t)=0.
In terms of Laplace transform, we have:
Y(s)=1s2(4s2+6s+5)=Cs+D4s(s+1)=0s4
where C=0 and D=0.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?