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

3kofbe 2021-11-23 Answered
Consider, ay+by+cy=0 and a0 Which of the following statements are always true?
1. A unique solution exists satisfying the initial conditions y(0)=π, y(0)=π
2. Every solution is differentiable on the interval (,)
3. If y1 and y2 are any two linearly independent solutions, then y=C1y1+C2y2 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=C1eλ1x+C2eλ2x
where, λ1 and λ2 are roots of the equation ax2+bx+c=0
1) if we know y(0) and y(0), then we can obtain two linear equations in C1 and C2, giving a unique solution.
2) As this function is exponential, we can say it is differentiable everywhere.
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Witheyesse47
Answered 2021-11-25 Author has 14 answers
the reason the (1) and (2) are correct is the nonsingular (a0)) 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 y1,y2 with y1(0)=1, y1(0)=0, y2(0)=0, y2(0)=1 so that they can take care of any initial conditions. like the you have y(0)=π, y(0)=π
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