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Chapter 4: Q17E (page 164)

In Problems 13鈥�20, solve the given initial value problem.

y"-6y'+9y=0:y(0)=2,y'(0)=253

Short Answer

Expert verified

The solution is y(t)=2e3t+73te3t.

Step by step solution

01

Find the solution of the differential equation.

The given differential equation is y" -6y' + 9y = 0.

The auxiliary equation is;

r2-6r1+9r0=0r2-6r+9=0r2-3r-3r+9=0r(r-3)-3(r-3)=0(r-3)2=0 r=3,3

Therefore, the solution is y(t) = c1e3t + c2te3t.

02

Apply initial conditions.

The initial conditions are y(0)=2,y'(0)=253.

Therefore,

y(0)=c1e0+c2(0)e0c1=2

And

y'(t)=3c1e3t+c2(3te3t+e3t)y'(0)=3c1e0+c2(0+e0)3c1+c2=253

Solving for c1,c2 then;

c1=2c2=73

Therefore, the solution is y(t)=2e3t+73te3t.

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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.

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.y''-2y'+3y=cosht+sin3t

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation.2y''(x)-6y'(x)+y(x)=sinxe4x

Find a general solution y''-3y'-11y=0

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. y''=2y+2tan3x,鈥夆赌夆赌夆赌夆�夆赌�yp(x)=tanx
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