Chapter 7: Q15E (page 404)

Solve the given integral equation or integro-differential equation for y(t) $y (t) + 3 \int_{0}^{t} y (v) s i n (t - v) d v = t$

Short Answer

Expert verified

Integral equation or integro-differential equation for y(t) is . $\frac{1}{4} t + \frac{3}{8} \sin 2 t$

Step by step solution

Given the integral equation is,

$\begin{matrix} y (t) + 3 \int_{0}^{t} y (v) \sin (t - v) d v = t \\ y (t) + 3 [y (t) ⋆ \sin t] = t \end{matrix}$ [Using $g ⋆ h = \int_{0}^{t} g (t - v) h (v) d v$

Taking Laplace transform on both sides,

$\begin{matrix} y (s) + 3 [y (s) \cdot \frac{1}{s^{2} + 1}] = \frac{1}{s^{2}} \\ [1 + \frac{3}{s^{2} + 1}] y (s) = \frac{1}{s^{2}} \\ y (s) = \frac{s^{2} + 1}{s^{2} (s^{2} + 4)} \dots \dots (1) \end{matrix}$

Step 3: Using the partial function, we get

$\frac{s^{2} + 1}{s^{2} (s^{2} + 4)} = \frac{1}{4 s^{2}} + \frac{3}{4 (s^{2} + 4)}$

Now, equation (1) becomes

$\begin{matrix} y (s) = \frac{1}{4 s^{2}} + \frac{3}{4 (s^{2} + 4)} \\ = \frac{1}{4 s^{2}} + \frac{3}{8} \frac{2}{(s^{2} + 4)} \end{matrix}$

Taking inverse Laplace transform, we get

$y (t) = \frac{1}{4} t + \frac{3}{8} \sin 2 t$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Solve the given integral equation or integro-differential equation for y(t) $y (t) + 3 \int_{0}^{t} y (v) s i n (t - v) d v = t$

Short Answer

Step by step solution

Given the integral equation is,

Taking Laplace transform on both sides,

Step 3: Using the partial function, we get

Taking inverse Laplace transform, we get

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Probability and Statistics

Pure Maths

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Solve the given integral equation or integro-differential equation for y(t) y(t)+3∫0ty(v)sin(t−v)dv=t

Short Answer

Step by step solution

Given the integral equation is,

Taking Laplace transform on both sides,

Step 3: Using the partial function, we get

Taking inverse Laplace transform, we get

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Probability and Statistics

Pure Maths

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Solve the given integral equation or integro-differential equation for y(t) $y (t) + 3 \int_{0}^{t} y (v) s i n (t - v) d v = t$