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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: 19E (page 350)

In Problems $15 - 24$ , solve for , the Laplace transform of the solution $y (t)$ to the given initial value problem.

Short Answer

Expert verified

The Initial value for $y'' + 3 y = t^{3}$ is $Y (s) = \frac{6}{s^{4} (s^{2} + 3)}$

Step by step solution

01

Determine the Laplace Transform

The Laplace transform is a strong integral transform used in mathematics to convert a function from the time domain to the s-domain.
In some circumstances, the Laplace transform can be utilized to solve linear differential equations with given initial conditions.
$F (s) = {鈭�}_{0}^{鈭�} f (t) e^{- st} t'$

02

Determine the Laplace transform

Applying the Laplace transform and using its linearity we get

$L y'' + 3 y = L \{t^{3}\} L \{y''\} + 3 L\} = \frac{3!}{s^{4}}$

Solve for the Laplace transform as:

$[s^{2} Y (0) - s y (0) - y' (0)] + 3 Y (s) = \frac{6}{s^{4}} s^{2} Y (s) + 3 Y (s) = \frac{6}{s^{4}} (s^{2} + 3) Y (s) = \frac{6}{s^{4}} Y (s) = \frac{6}{s^{4} (s^{2} + 3)}$

Therefore, the initial value for $y'' + 3 y = t^{3}$ is $Y (s) = \frac{6}{s^{4} (s^{2} + 3)}$

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Most popular questions from this chapter

In Problems 3-10, determine the Laplace transform of the given function. $t^{2} u (t - 4)$

solve the given initial value problem using the method of Laplace transforms. Sketch the graph of the solution.

$y'' + y = u (t - - 3); y (0) = 0, y' (0) = 1$

Determine the inverse Laplace transform of the given function.

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

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

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

$y'' + 4 y = 4 t^{2} - 4 t + 10; y (0) = 0, y' (0) = 3$

See all solutions

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