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Chapter 4: Q7E (page 164)

Question: Find a general solution to the given differential equation.

6y''+y'-2y=0

Short Answer

Expert verified

Answer

The general solution of the given equation isy=c1e12t+c2e-23t.

Step by step solution

01

Write the auxiliary equation of the given differential equation.

The given differential equation is6y''+y'-2y=0.

The auxiliary equation for the above equation6m2+m-2=0.

02

Now find the roots of the auxiliary equation. 

Solve the auxiliary equation,

6m2+m-2=06m2+4m-3m-2=02m3m+2-13m+2=03m+22m-1=0

The roots of the auxiliary equation are

m1=12, & m2=-23.

03

Write the general solution.

If an auxiliary equation has distinct real roots as&, then the general solution is given as;

y=c1em1t+c2em2t

Thus, the general solution of the given equation is

y=c1e12t+c2e-23t.

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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);    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 the solution to the initial value problem.

y''(x)-y'(x)-2y(x)=cosx-sin2x;     y(0)=-720,    y'(0)=15

Find a general solution to the differential equation.y''(x)+6y'(x)+10y(x)=10x4+24x3+2x2-12x+18

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution to the given equation. y''+2y'-y=t-1et

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