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Chapter 4: Q2RP (page 231)

Find a general solution to the given differential equation.49y''+14y'+y=0

Short Answer

Expert verified

y=c1e-17t+c2te-17t

Step by step solution

01

Write the auxiliary equation of the given differential equation

The given differential equation is49y''+14y'+y=0.

The auxiliary equation for the above equation49m2+14m+1=0.

02

Find the roots of the auxiliary equation  

Solve the auxiliary equation,

49m2+14m+1=049m2+7m+7m+1=07m7m+1+17m+1=07m+17m+1=0

The roots of the auxiliary equation arem1=-17鈥�&鈥�m2=-17.

Thus, the general solution of the given equation isy=c1e-17t+c2te-17t.

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Most popular questions from this chapter

Find a particular solution to the differential equation.

d2ydx2-5dydx+6y=xex

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) y''-6y'+9y=5t6e3t

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.

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

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

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation.y''+5y'+6y=6x2+10x+2+12ex,鈥夆赌夆赌夆赌夆�夆赌�yp(x)=ex+x2
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