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Consider, $ay'' + by' + cy = 0$ and $a \ne 0.$ Which of the following statements are always true?

  1. A unique solution exists satisfying the initial conditions $y(0) = \pi, y'(0) = sqrt(\pi).$

  2. Every solution is differentiable on the interval $(-\infty, \infty).$

  3. If $y_1$ and $y_2$ are any two linearly independent solutions, then $y = C_1y_1 + C_2y_2$ is a general solution of the equation.

The answer is all of the above. I understand why 3) is correct, but how can one know for certain that 1) and 2) are correct statements?

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2 Answers 2

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The general solution to this equation is, $$y=C_1e^{\lambda_1x}+C_2e^{\lambda_2x}$$ where,$\lambda_1$ and $\lambda_2$ are roots of the equation $ax^2+bx+c=0$

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.

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the reason the $(1)$ and (2) are correct is the nonsingular($a \ne 0$)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 $y(0) = \pi, y'(0) = \sqrt \pi.$

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