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Chapter 4: Q42E (page 200)

A differential equation and a nontrivial solution fare given. Find a second linearly independent solution using reduction of order.

t2y''+6ty'+6y=0,t>0;f(t)=t-2

Short Answer

Expert verified

The second linearly independent solution of the given equation

t2y''+6ty'+6y=0,t>0;f(t)=t-2is y=c1t-2+c2t-3.

Step by step solution

01

Substitute the values

Given differential equation is t2y''+6ty'+6y=0then the standard form of the solution is y''+6ty'+6t2y=0and f(t)=t-2is one of the solutions and p(t)=6t. Then

y2=y1(t)鈭�e-鈭�p(t)dty12(t)dt=t-2鈭�e-鈭�6tdtt-22dt

02

Simplification

Simplify the above and get;

=t-2鈭�elnt-6t-4=t-2脳鈭�t-6t-4=t-2脳-t-1=-t-3

So, the solution is y=c1t-2+c2t-3.

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Most popular questions from this chapter

Find a particular solution to the differential equation.y''+3y=-9

Find a general solution to the differential equation.

y''+4y=蝉颈苍胃-肠辞蝉胃

Find the solution to the initial value problem.

y''=6t;鈥夆赌夆赌夆赌夆赌�y(0)=3,鈥夆赌夆赌夆赌�y'(0)=-1

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y''(x)-3y'(x)+2y(x)=exsinx

Vibrating Spring without Damping. A vibrating spring without damping can be modeled by the initial value problem(11)in Example3 by taking b=0.

a) If m=10kg,k=250kg/sec2,y(0)=0.3m, and y'(0)=-0.1m/sec, find the equation of motion for this undamped vibrating spring.

b)After how many seconds will the mass in part (a) first cross the equilibrium point?

c)When the equation of motion is of the form displayed in (9), the motion is said to be oscillatory with frequency 尾/2蟺. Find the frequency of oscillation for the spring system of part (a).
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