Chapter 4: Q29E (page 200)

Prove that if $y_{1}$ and $y_{2}$ are linearly independent solutions of $y'' + py' + qy = 0$ on $(a, b)$ , then they cannot both be zero at the same point $t_{0}$ in $(a, b)$

Short Answer

Expert verified

$y_{1}$ and $y_{2}$ cannot be zero at the same point $t_{0}$ in $(a, b)$ .

Step by step solution

Check linear independence.

Let $y_{1}$ and $y_{2}$ are two independent solutions of the given differential equations, then $c_{1} y_{1} + c_{2} y_{2} = 0$ only if $c_{1} = c_{2} = 0$ .

Check whether y1,y2can be zero or not.

If $y_{1} (t_{0}) = y_{2} (t_{0}) = 0$ , then $c_{1} y_{1} (t_{0}) + c_{2} y (t_{0}) = 0$ even if $c_{1} = c_{2} \neq 0$ .

Thus, it contradicts the linear independence of the solutions. Therefore $y_{1}$ and $y_{2}$ cannot be zero at the same point.

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Prove that if $y_{1}$ and $y_{2}$ are linearly independent solutions of $y'' + py' + qy = 0$ on $(a, b)$ , then they cannot both be zero at the same point $t_{0}$ in $(a, b)$

Short Answer

Step by step solution

Check linear independence.

Check whether y1,y2can be zero or not.

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Most popular questions from this chapter

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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

Step by step solution

Check linear independence.

Check whether y1,y2can be zero or not.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Prove that if $y_{1}$ and $y_{2}$ are linearly independent solutions of $y'' + py' + qy = 0$ on $(a, b)$ , then they cannot both be zero at the same point $t_{0}$ in $(a, b)$