Chapter 8: Q11P (page 443)

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.
$y " - 4 y = 4 e^{2 t}, \sum_{} y_{0} = 0, y'_{0} = 1$

Short Answer

Expert verified

The solution of given differential equation is y = $t e^{2 t}$ .

Step by step solution

Given information

The equation is $y " - 4 y = 4 e^{2 t} and y_{0} = 0, y'_{0} = 1$ .

Step 2: Definition of Laplace Transformation

A transformation of a function f(x) into the function g(t) that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation.

The inverse Laplace transform of a function F(s) is the piecewise-continuous and exponentially-restricted real function f(t)

Differentiate the given function

Consider the equation.

$y " - 4 y = 4 e^{2 t}$

Take the Laplace of above equation.

$L \{y " - 4 y\} = L \{4 e^{2 i}\} L \{y "\} - 4 L \{y\} = 4 L \{e^{2 t}\} p^{2} L \{y\} - p y_{0} - y'_{0} - 4 L \{y\} = \frac{4}{p - 2} (p^{2} - 4) L \{y\} - p y_{0} - y'_{0} = \frac{4}{p - 2}$

Further solve,

$(p^{2} - 4) L \{y\} - p (0) - 1 = \frac{4}{p - 2} (p^{2} - 4) L \{y\} = \frac{4}{p - 2} + 1 L \{y\} = \frac{4}{p - 2 (p^{2} - 4)} + \frac{1}{(p^{2} - 4)} L \{y\}$

The inverse Laplace is,

$y = L^{- 1} \{{(p - 2)}^{2}\} y = t e^{2 t}$

Thus, the solution of given differential equation is $y = t e^{2 t}$ .

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By using Laplace transforms, solve the following differential equations subject to the given initial conditions.
$y " - 4 y = 4 e^{2 t}, \sum_{} y_{0} = 0, y'_{0} = 1$

Short Answer

Step by step solution

Given information

Step 2: Definition of Laplace Transformation

Differentiate the given function

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By using Laplace transforms, solve the following differential equations subject to the given initial conditions.y"-4y=4e2t,∑y0=0,y'0=1

Short Answer

Step by step solution

Given information

Step 2: Definition of Laplace Transformation

Differentiate the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Physics of Motion

Dynamics

Kinematics Physics

Magnetism

Electromagnetism

Study anywhere. Anytime. Across all devices.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.
$y " - 4 y = 4 e^{2 t}, \sum_{} y_{0} = 0, y'_{0} = 1$