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Chapter 7: Q16E (page 404)

Solve the given integral equation or integro-differential equation for y(t)y(t)+3鈭�0tet鈭�vy(v)dv=sint

Short Answer

Expert verified

The integral equation or integro-differential equation for y(t)is.cost+sint鈭�1

Step by step solution

01

Given the integral equation is,

y(t)+鈭�0tet鈭�vy(v)dv=sinty(t)+[et鈰�y(t)]=sint [Using g鈰�h=鈭�0tg(t鈭�v)h(v)dv

02

Taking Laplace transform on both sides, we get

y(s)+[y(s)鈰�1s鈭�1]=1s2+1[1+1s鈭�1]y(s)=1s2+1y(s)=s鈭�1s(s2+1)鈥�鈥�(1)

03

Step 3: Using the partial function, we get

s鈭�1s(s2+1)=s+1s2+1鈭�1s

Hence, the first equation becomes

y(s)=s+1s2+1鈭�1s=ss2+1+1s2+1鈭�1s

04

Step 3: Taking inverse Laplace transform, we get  

y(t)=cost+sint鈭�1

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Most popular questions from this chapter

In Problems 3-10, determine the Laplace transform of the given function.f(t)=cost,-蟺/2鈮�t鈮�蟺/2and f(t)has period蟺

In Problems 11鈥�20, determine the partial fraction expansion for the given rational function.鈭�2s2鈭�3s鈭�2s(s+1)2

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.]


cos3t

In Problems 1-14, solve the given initial value problem using the method of Laplace transforms.

y''+y=t;y蟺=0,y'蟺=0

In Problems 1 and 2, use the definition of the Laplace transform to determine L{f}.

f(t)={e-t,0鈮�t鈮�5-1,5<t
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