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Chapter 4: Q28E (page 164)

In Problems 27鈥�32, use Definition 1 to determine whether the functions y1and y2are linearly dependent on the interval (0, 1).

28. y1(t) = e3t, y2(t) = e-4t

Short Answer

Expert verified

The functions are not linearly dependent.

Step by step solution

01

Apply the given values.

The given functions are y1(t) = e3t, y2(t) = e-4t

If the y1 and y2 are linearly dependent on the interval (0, 1) then

y1(t) = C1y2(t)

02

Find linearly dependent functions.

Now,

e3t鈮�Ce-4t

This cannot be expressed as e3t multiples of e-4t.

Therefore, these functions are not linear dependent.

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Most popular questions from this chapter

Discontinuous Forcing Term. In certain physical models, the nonhomogeneous term, or forcing term, g(t) in the equation

ay''+by'+cy=g(t)

may not be continuous but may have a jump discontinuity. If this occurs, we can still obtain a reasonable solution using the following procedure. Consider the initial value problem;

y''+2y'+5y=g(t);鈥夆赌夆赌夆赌�y(0)=0,鈥夆赌夆赌夆赌�y'(0)=0

Where,

g(t)=10,鈥夆赌�if鈥�0鈮�t鈮�3蟺20,鈥夆赌夆赌夆赌夆��if鈥�t>3蟺2

  1. Find a solution to the initial value problem for 0鈮�t鈮�3蟺2 .
  2. Find a general solution fort>3蟺2.
  3. Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first derivatives, att=3蟺2 . This gives us a continuously differentiable function that satisfies the differential equation except at t=3蟺2.

Find a general solution to the differential equation.

y''-y=-11t+1

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. ty''-y'+2y=sin3t

Solve the given initial value problem. w''-4w'+2w=0;w(0)=0,w'(0)=1

The auxiliary equation for the given differential equation has complex roots. Find a general solution.y''-10y'+26y=0
See all solutions

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