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Chapter 4: Q29E (page 200)

Prove that if y1andy2are linearly independent solutions ofy''+py'+qy=0on(a,b), then they cannot both be zero at the same pointt0in(a,b)

Short Answer

Expert verified

y1andy2 cannot be zero at the same pointt0 in a,b.

Step by step solution

01

Check linear independence. 

Let y1andy2 are two independent solutions of the given differential equations, then c1y1+c2y2=0only if c1=c2=0.

02

Check whether y1,y2can be zero or not.

If y1(t0)=y2(t0)=0, then c1y1(t0)+c2y(t0)=0even if c1=c2鈮�0.

Thus, it contradicts the linear independence of the solutions. Therefore y1 andy2 cannot be zero at the same point.

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Most popular questions from this chapter

Find the solution to the initial value problem.

z''(x)+z(x)=2e-x,鈥夆赌夆赌夆赌夆赌夆赌�z(0)=0,鈥夆赌夆赌夆赌夆赌夆赌夆��z'(0)=0

Find a particular solution to the differential equation.

2z''+z=9e2t

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

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.y''-2y'+3y=cosht+sin3t

Given that y1(t)=costis a solution to y''-y'+y=sintand y2(t)=e2t3is a solution to role="math" localid="1654926813168" y''-y'+y=e2t. Use the superposition principle to find solutions to the following differential equations:

(a)鈥夆赌夆赌夆赌�y''-y'+y=5sint

(b)鈥夆赌夆赌夆赌�y''-y'+y=sint-3e2t

(c)鈥夆赌夆赌夆赌�y''-y'+y=4sint+18e2t
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