Q15E Solve the given integral equatio... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 7: Q15E (page 404)

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

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,

$\begin{matrix} y (t) + 3 {鈭�}_{0}^{t} y (v) \sin (t 鈭� v) d v = t \\ y (t) + 3 [y (t) 鈰� \sin t] = t \end{matrix}$ [Using $g 鈰� h = {鈭�}_{0}^{t} g (t 鈭� v) h (v) d v$

Taking Laplace transform on both sides,

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

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

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

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91影视

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

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

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91影视

Solve the given integral equation or integro-differential equation for y(t) y(t)+3鈭�0ty(v)sin(t鈭�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

Mechanics Maths

Decision Maths

Statistics

Applied Mathematics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

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