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Chapter 4: Q8E (page 180)

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

Short Answer

Expert verified

No, the method of undetermined coefficients can’t be applied.

Step by step solution

01

Use the method of undetermined coefficients to find a particular solution of a given differential equation.

Given equation isty''-y'+2y=sin3t.

Here, the given differential equation is non-homogeneous.

According to the method of undetermined coefficients, the method of undetermined coefficients applies to the differential equation of the following form,

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

Here, a, b and c are the constants.

02

Final conclusion.

The coefficient ofy'' is twhich is not constant. Therefore, we can say that L.H.S. of the given equation is not a differential equation with a constant coefficient.

Thus, the method of undetermined coefficients cannot be applied.

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Most popular questions from this chapter

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

Find a particular solution to the differential equation.

θ''(t)-θ(t)=tsint

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)

y''+3y'-7y=t4et

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);    y(0)=0,    y'(0)=0

Where,

g(t)=10,  if 0≤t≤3π20,     if t>3π2

  1. Find a solution to the initial value problem for 0≤t≤3π2 .
  2. Find a general solution fort>3π2.
  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=3π2 . This gives us a continuously differentiable function that satisfies the differential equation except at t=3π2.

Find a particular solution to the given higher-order equation.y4-3y'''+3y''-y'=6t-20
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