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Chapter 1: 20E (page 1)

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

y''+3y=t3;y0=0,y'0=0

Short Answer

Expert verified

The Initial value fory''+3y=t3forYs=6s4s2+3

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 initial conditions.
  • Fs=∫0∞f(t)e-stt'
02

Determine the Laplace transform

Applying the Laplace transform and using its linearity we get

Ly''+3y=Lt3Ly''+3L=3!s4

Solve for the Laplace transform as:

s2Y0-sy0-y'0+3Ys=6s4s2Ys+3Ys=6s4s2+3Ys=6s4Ys=6s4s2+3

Therefore, the initial value fory''+3y=t3 isYs=6s4s2+3

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Most popular questions from this chapter

In Problems 10–13, use the vectorized Euler method with h = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

y''=t2-y2;y(0)=0,y'(0)=1on[0,1]

In Problems 3–8, determine whether the given function is a solution to the given differential equation.

y=e2x-3e-x,d2ydx2-dydx-2y=0

In Problems 14–24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge–Kutta algorithm. (At the instructor’s discretion, other algorithms may be used.)†

Using the vectorized Runge–Kutta algorithm, approximate the solution to the initial value problem y''=t2+y2;y(0)=1,y'(0)=0at t=1. Starting with h=1, continue halving the step size until two successive approximations of both y1andy'1differ by at most 0.1.

Consider the differential equation dydx=x+siny

⦁ A solution curve passes through the point (1,π2). What is its slope at this point?

⦁ Argue that every solution curve is increasing for x>1.

⦁ Show that the second derivative of every solution satisfies d2ydx2=1+xcosy+12sin2y.

⦁ A solution curve passes through (0,0). Prove that this curve has a relative minimum at (0,0).

In Problems 14–24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge–Kutta algorithm. (At the instructor’s discretion, other algorithms may be used.)†

Using the vectorized Runge–Kutta algorithm, approximate the solution to the initial value problem

dudx=3u-4v;u(0)=1'dvdx=2u-3v;v(0)=1

at x = 1. Starting with h=1, continue halving the step size until two successive approximations of u(1)and v(1) differ by at most 0.001.
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