Chapter 8: Q2P (page 442)

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $y^{t} - y = 2 t^{t}, 2 t = 3$

Short Answer

Expert verified

Answer

The solution of given differential equation is $y - e^{t} (3 + 2 t)$ .

Step by step solution

Given information

The given equation is $y - y = 2 e^{t}$ and $y_{0} - 3$ .

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

$y^{'} - y = 2 e^{x}$

Take the Laplace of above equation.

$L \{y - y\} = L \{2,\} n L \{y'\} - L \{y\} = L \{2 e\}$

Use $L \{e^{w}\} = \frac{1}{p + a}$ and $L \{y'\} = p L \{y\} - y$ in above equation.

$p L \{y\} - y_{0} - L \{y\} = \frac{2}{p - 1} p L \{y\} - 3 - L \{y\} = \frac{2}{p - 1} L \{y\} (p - 1) = \frac{3 p - 1}{p - 1} L \{y\} = \frac{3 p - 1}{{(p - 1)}^{2}}$

The inverse Laplace is,

$y = L^{- 1} \{\frac{3 p - 1}{{(p - 1)}^{2}}\} = 3 L^{- 1} \{\frac{p}{{(p - 1)}^{2}}\} - L^{- 1} \{\frac{1}{{(p - 1)}^{2}}\} = 3 e^{2} (1 + t) - t e^{2} y = e' (3 + 2 t)$

Thus, the solution of given differential equation is $y = e^{t} (3 + 2 t)$ .

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By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $y^{t} - y = 2 t^{t}, 2 t = 3$

Short Answer

Step by step solution

Given information

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.yt-y=2tt,2t=3

Short Answer

Step by step solution

Given information

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

Turning Points in Physics

Mechanics and Materials

Physics of Motion

Energy Physics

Fields in Physics

Wave Optics

Study anywhere. Anytime. Across all devices.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $y^{t} - y = 2 t^{t}, 2 t = 3$