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Chapter 4: Q9E (page 164)

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

4y''-4y'+y=0

Short Answer

Expert verified

Answer

The general solution of the given equation isy=c1+c2te12t.

Step by step solution

01

Write the auxiliary equation of the given differential equation.

The given differential equation is

4y''-4y'+y=0.

The auxiliary equation for the above equation

4m2-4m+1=0.

02

Now find the roots of the auxiliary equation. 

Solve the auxiliary equation,

4m2-4m+1=02m2-221m+1=02m-12=0

The roots of the auxiliary equation arem1=12,鈥�&鈥�m2=12.

03

Write the general solution.

If the auxiliary equation has repeated real roots, then the general solution is given as;

y=c1+c2temt

Thus, the general solution of the given equation isy=c1+c2te12t.

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Most popular questions from this chapter

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

Find a particular solution to the differential equation.

y''+2y'-y=10

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

The auxiliary equations for the following differential equations have repeated complex roots. Adapt the "repeated root" procedure of Section 4.2 to find their general solutions:

(a)y''''+2y''+y=0

(b)y''''+4y'''+12y''+16y'+16y=0

Find a general solution to the differential equation.

y''-y=-11t+1
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