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Chapter 7: Q17E (page 404)

In problem 15-22, solve the given integral equation or integro differential equation fory(t)

L鈭�0t(t鈭�v)y(v)dv=1

Short Answer

Expert verified

The integral equation or integro-differential equation for y(t)is.y(t)=cost

Step by step solution

01

Definition of Laplace transform

A transformation of a function f(x) into the function is particularly useful for reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation.

Applying the Laplace transform we get

L{y(t)+鈭�0t(t鈭�v)y(v)dv}(s)={1}L{y(t)}(s)+L{鈭�0t(t鈭�v)y(v)dv}(s)=1s

Y(s)=L{t*y(t)}(s)=1s

Y(s)=L{t}(s).L{y(t)}=1s

Y(s)=1+1s2Y(s)=1s

(1+1s2)Y(s)=1sY(s)=sS(s2+1)

02

Definition of inverse

Now,

using inverse Laplace we get

y(t)=L鈭�1{sS(s2+1)}=cost

Hence,

y(t)=cost

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Most popular questions from this chapter

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]


sin2tsin5t

Determine the inverse Laplace transform of the given function.


s+1s2+2s+10.

Use the method of Laplace transforms to solve

x''+y'=2;x(0)=3,x'(0)=04x+y'=6;y(1)=4.

Determine the inverse Laplace transform of the given function.

1s2+4s+8.

Determine the inverse Laplace transform of the given function.

2s2+3s-1(s+1)2(s+2)
See all solutions

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