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Chapter 7: 17E (page 350)

In Problems 15-24, solve for Y(s), the Laplace transform of the solution y(t) to the given initial value problem.

17.y''+y'-y=t3;y(0)=1,y'(0)=0

Short Answer

Expert verified

The Initial value fory''+y'-y=t3isY=s5+s4+6s4s2+s-1

Step by step solution

01

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.
  • Fs=鈭�0鈭�f(t)e-stt'
02

Determine the Laplace transform

Define Lys=Ys

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

Ly's=sLys-y0Ly''s=s2Lys-sy0-y'0Ctns=n!sn+1Ly''+Ly'-Ly=Lt3

Substitute the properties into the equation.

s2Y-sy(0)-y'(0)+[sY-y(0)]-Y=3!s4

Substitute the initial conditions

y0=1andy'0=0s2Y-s+sY-1-Y=6s4

Isolate the Y variable and solve:

s2Y+sY-Y=6s4+s+1Ys2+s-1=s5+s4+6s4Y=s5+s4+6s4s2+s-1

Therefore, the initial value fory''+y'-y=t3isY=s5+s4+6s4s2+s-1

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Most popular questions from this chapter

The transfer function of a linear system is defined as the ratio of the Laplace transform of the output function y(t) to the Laplace transform of the input function g(t), when all initial conditions are zero. If a linear system is governed by the differential equation


y''(t)+6y'(t)+10y(t)=g(t),t>0

use the linearity property of the Laplace transform and Theorem 5 on page363 on the Laplace transform of higher-order derivatives to determine the transfer function H(s)=Y(s)/G(s)of this system.

In Problems 1-20, determine the Laplace transform of the given function using Table 7.1 on page 356 and the properties of the transform given in Table 7.2. [Hint: In Problems 12-20, use an appropriate trigonometric identity.]


cosntsinmt,m鈮�n

Determine the inverse Laplace transform of the given function.

2s2+3s-1(s+1)2(s+2)

In Problems 1-14, solve the given initial value problem using the method of Laplace transforms

10.y''-4y=4t-8e-2t;y(0)=0,y'(0)=5

Given that L{cosbt}(s)=s/(s2+b2), use the translation property to compute L{eatcosbt}.
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