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Chapter 8: Q2P (page 442)

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.yt-y=2tt,2t=3

Short Answer

Expert verified

Answer

The solution of given differential equation isy-et3+2t.

Step by step solution

01

Given information

The given equation is y-y=2etandy0-3.

02

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)

03

Differentiate the given function

Consider the equation

y'-y=2ex

Take the Laplace of above equation.

Ly-y=L2,nLy'-Ly=L2e

Use Lew=1p+aand Ly'=pLy-yin above equation.

pLy-y0-Ly=2p-1pLy-3-Ly=2p-1Lyp-1=3p-1p-1Ly=3p-1p-12

The inverse Laplace is,

y=L-13p-1p-12=3L-1pp-12-L-11p-12=3e21+t-te2y=e'3+2t

Thus, the solution of given differential equation isy=et3+2t.

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Most popular questions from this chapter

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves.

2. x1-y2dx+y1-x2dy=0,y=12whenx=12

Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from y'for the original curves; this constant takes different values for different curves of the original family, and you want an expression for y'which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations 2.10to 2.12.

(y-1)2=x2+k

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves.

11.2y'=3(y-2)13 y = 3when x = 1

Suppose the rate at which bacteria in a culture grow is proportional to the number present at any time. Write and solve the differential equation for the number N of bacteria as a function of time t if there are N0bacteria when t=0. Again note that (except for a change of sign) this is the same differential equation and solution as in the preceding problems.

Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from y'for the original curves; this constant takes different values for different curves of the original family, and you want an expression for y'which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations (2.10)to (2.12)

x2+y2=cost.
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