Solve y" + y = tet, y(0) = 0, y'(0) = 0 using laplace transforms.

The solution of the diferential equation is:

Step-by-step explanation:

Given ; y(0) = 0 ; y'(0) = 0

We need to use the Laplace transform to solve it.

ℒ[y" + y]=ℒ[]

ℒ[y"]+ℒ[y]=ℒ[]

By using the Table of Laplace Transform we get:

ℒ[y"]=s²·ℒ[y]+s·y(0)-y'(0)=s²·Y(s)

ℒ[y]=Y(s)

ℒ[]=

So, the transformation is equal to:

s²·Y(s)+Y(s)=

(s²+1)·Y(s)=

Y(s)=

To be able to separate in terms, we use the partial fraction method:

1=(As+B)(s-1)² + C(s-1)(s²+1)+ D(s²+1)

The equation is reduced to:

1=s³(A+C)+s²(B-2A-C+D)+s(A-2B+C)+(B+D-C)

With the previous equation we can make an equation system of 4 variables.

The system is given by:

A+C=0

B-2A-C+D=0

A-2B+C=0

B+D-C=1

The solution of the system is:

A=1/2 ; B=0 ; C=-1/2 ; D=1/2

Therefore, Y(s) is equal to:

Y(s)=

By using the inverse of the Laplace transform:

ℒ⁻¹[Y(s)]=ℒ⁻¹[]-ℒ⁻¹[]+ℒ⁻¹[]

120 seems like the most sensible answer.

step-by-step explanation:

if there is only one topping on all caramel apples and there's 120 of them. then all apples should/will be covered in sprinkles. make sense?

at least two

step-by-step explanation: