Chapter 7: 18E (page 350)

In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' - 2 y' - y = e^{2 t} - e^{t}; y (0) = 1, y' (0) = 3$

Short Answer

Expert verified

The Initial value for $y'' - 2 y' - y = e^{2 t} - e^{t}$ is $Y (s) = \frac{s^{3} - 2 s^{2} - s + 3}{(s^{2} - 2 s - 1) (s - 2) (s - 1)}$

Step by step solution

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 beginning conditions.
$F (s) = \int_{0}^{\infty} f (t) e^{- st} t'$

Determine the Laplace transform

Applying the Laplace transform and using its linearity we get

$L \{y'' - 2 y' - y\} = L \{e^{2 t} - e^{t}\} L \{y''\} - 2 L \{y'\} - L \{y\} = \frac{1}{s - 2} - \frac{1}{s - 1}$

Solve for the Laplace transform as:

$[s^{2} Y (s) - s y (0) - y' (0)] - 2 [s Y (s) - y (0)] - Y (s) = \frac{1}{(s - 2) (s - 1)} s^{2} Y (s) - s - 3 - 2 s Y (s) + 2 - Y (s) = \frac{1}{(s - 2) (s - 1)}$

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

Solve further as:

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

Therefore, the Initial value for $y'' - 2 y' - y = e^{2 t} - e^{t}$ is $Y (s) = \frac{s^{3} - 2 s^{2} - s + 3}{(s^{2} - 2 s - 1) (s - 2) (s - 1)}$

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In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' - 2 y' - y = e^{2 t} - e^{t}; y (0) = 1, y' (0) = 3$

Short Answer

Step by step solution

Determine the Laplace Transform

Determine the Laplace transform

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In Problems 15-24 , solve for Ys , the Laplace transform of the solution y(t) to the given initial value problem.y''-2y'-y=e2t-et;y0=1,y'0=3

Short Answer

Step by step solution

Determine the Laplace Transform

Determine the Laplace transform

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Pure Maths

Calculus

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' - 2 y' - y = e^{2 t} - e^{t}; y (0) = 1, y' (0) = 3$