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Chapter 1: Q5E (page 1)

Find a general solution for the differential equation with x as the independent variable:

ym+3yn+28y'+26y=0

Short Answer

Expert verified

The general solution for the differential equation with x as the independent variable is y=c1e−x+c2e−xcos5x+c3e−xsin5x

Step by step solution

01

Auxiliary equation:

The auxiliary equation for y'''+3y''+28y'+26=0is r3+3r2+28r+26=0. The solution of previous equation is:

r3+3r2+28r+26=0(r+1)(r2+2r+26)=0r=−1r2+2r+26=0r2,3=−2±22−4.262=−1±5i

Where I is imaginary unit

02

General solution:

Then α=−1,β=5 and we can write the general solution for the differential equation:

y=c1e−x+c2e−xcos5x+c3e−xsin5x

Hence the final solution is y=c1e−x+c2e−xcos5x+c3e−xsin5x

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Most popular questions from this chapter

Decide whether the statement made is True or False. The relation x2+y3-ey=1 is an implicit solution to dydx=ey-2x3y2.

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

y-logey=x2+1,dydx=2xyy-1

Spring Pendulum.Let a mass be attached to one end of a spring with spring constant kand the other end attached to the ceiling. Letlo be the natural length of the spring, and let l(t) be its length at time t. Ifθ(t) is the angle between the pendulum and the vertical, then the motion of the spring pendulum is governed by the system

l''(t)-l(t)θ'(t)-gcosθ(t)+km(l-lo)=0l2(t)θ''(t)+2l(t)l'(t)θ'(t)+gl(t)sinθ(t)=0

Assume g = 1, k = m = 1, and lo= 4. When the system is at rest, l=lo+mgk=5.

a. Describe the motion of the pendulum when l(0)=5.5,l'(0)=0,θ(0)=0,θ'(0)=0.

b. When the pendulum is both stretched and given an angular displacement, the motion of the pendulum is more complicated. Using the Runge–Kutta algorithm for systems with h = 0.1 to approximate the solution, sketch the graphs of the length l and the angular displacement u on the interval [0,10] if l(0)=5.5,l'(0)=0,θ(0)=0.5,θ'(0)=0.

In problems 1-6, identify the independent variable, dependent variable, and determine whether the equation is linear or nonlinear.

6-t3(d2vdt2)+3v-lnt=dvdt

In Problems 9-13, determine whether the given relation is an implicit solution to the given differential equation. Assume that the relationship implicitly defines y as a function of x and use implicit differentiation.

siny+xy-x3=2,y''=6xy'+y'3siny-2y'23x2-y
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