Chapter 7: 16E (page 350)

Question: In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' + 6 y = t^{2} - 1; y (0) = 0, y' (0) = - 1$

Short Answer

Expert verified

The Initial value for $y'' + 6 y = t^{2} - 1$ is $Y = \frac{- s^{3} - s^{2} + 2}{s^{3} (s^{2} + 6)}$

Step by step solution

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

Define $L \{y\} (s) = Y (s)$

Using the properties listed below, take the Laplace transform of the equation.

$L \{y''\} (s) = s^{2} L \{y\} (s) - s y (0) - y' (0) L \{t^{n}\} (s) = \frac{n!}{s^{n + 1}}$

$L \{1\} (s) = \frac{1}{s} L \{y''\} + 6 L \{y\} = L \{t^{2}\} - L \{1\}$

Substitute the properties into the equation

$[s^{2} Y - s y (0) - y' (0)] + 6 Y = \frac{2!}{s^{3}} - \frac{1}{s}$

Substitute the initial conditions

$y (0) = 0 a n d y' (0) = - 1$

$(s^{2} Y + 1) + 6 Y = \frac{2}{s^{3}} - \frac{1}{s}$

Isolate the Y variable as:

$Y (s^{2} + 6) = \frac{2}{s^{3}} - \frac{1}{s} - 1 = \frac{- s^{3} - s^{2} + 2}{s^{3}}$

$Y = \frac{- s^{3} - s^{2} + 2}{s^{3} (s^{2} + 6)}$

Therefore, the initial value for $y'' + 6 y = t^{2} - 1$ is $Y = \frac{- s^{3} - s^{2} + 2}{s^{3} (s^{2} + 6)}$

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Question: In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' + 6 y = t^{2} - 1; y (0) = 0, y' (0) = - 1$

Short Answer

Step by step solution

The Laplace Transform

Determine the Laplace transform

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Question: In Problems 15-24, solve for Ys, the Laplace transform of the solution yt to the given initial value problem.y''+6y=t2-1;y0=0,y'0=-1

Short Answer

Step by step solution

The Laplace Transform

Determine the Laplace transform

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Decision Maths

Logic and Functions

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

Question: In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.
$y'' + 6 y = t^{2} - 1; y (0) = 0, y' (0) = - 1$