Consider, ay''+by'+cy=0 and a\ne0 Which of the following statements are

Question

Consider, ay''+by'+cy=0 and a\ne0 Which of the following statements are

3kofbe 2021-11-23 Answered

Consider,

a y^{″} + b y^{'} + c y = 0

and

a \neq 0

Which of the following statements are always true?
1. A unique solution exists satisfying the initial conditions

y (0) = π, y^{'} (0) = \sqrt{π}

2. Every solution is differentiable on the interval

(- \infty, \infty)

3. If

y_{1}

and

y_{2}

are any two linearly independent solutions, then

y = C_{1} y_{1} + C_{2} y_{2}

is a general solution of the equation.

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Expert Answer

Thomas Conway

Answered 2021-11-24 Author has 10 answers

The general solution to this equation is,

y = C_{1} e^{λ_{1} x} + C_{2} e^{λ_{2} x}

where,

λ_{1}

and

λ_{2}

are roots of the equation

a x^{2} + b x + c = 0

1) if we know

y (0)

and

y^{'} (0)

, then we can obtain two linear equations in

C_{1}

and

C_{2}

, giving a unique solution.
2) As this function is exponential, we can say it is differentiable everywhere.

This is helpful 0

Witheyesse47

Answered 2021-11-25 Author has 14 answers

the reason the (1) and (2) are correct is the nonsingular

(a \neq 0)

) linear equations have the uniqueness and existence property. all

y, y^{'}, y^{″}

stay bounded in the finite part of the domain. it always has two linearly independent solutions

y_{1}, y_{2}

with

y_{1} (0) = 1, y_{1}^{'} (0) = 0, y_{2} (0) = 0, y_{2}^{'} (0) = 1

so that they can take care of any initial conditions. like the you have

y (0) = π, y^{'} (0) = \sqrt{π}

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Analysis

Consider, ay''+by'+cy=0 and a\ne0 Which of the following statements are

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