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

Consider, $ay{}^{″}+b{y}^{\prime }+cy=0$ and $a\ne 0$ Which of the following statements are always true?
1. A unique solution exists satisfying the initial conditions
2. Every solution is differentiable on the interval $\left(-\mathrm{\infty },\mathrm{\infty }\right)$
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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Thomas Conway
The general solution to this equation is,
$y={C}_{1}{e}^{{\lambda }_{1}x}+{C}_{2}{e}^{{\lambda }_{2}x}$
where, ${\lambda }_{1}$ and ${\lambda }_{2}$ are roots of the equation $a{x}^{2}+bx+c=0$
1) if we know $y\left(0\right)$ and ${y}^{\prime }\left(0\right)$, 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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Witheyesse47
the reason the (1) and (2) are correct is the nonsingular $\left(a\ne 0\right)$) linear equations have the uniqueness and existence property. all $y,{y}^{\prime },y{}^{″}$ stay bounded in the finite part of the domain. it always has two linearly independent solutions ${y}_{1},{y}_{2}$ with so that they can take care of any initial conditions. like the you have