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

Consider, $$\displaystyle{a}{y}{''}+{b}{y}'+{c}{y}={0}$$ and $$\displaystyle{a}\ne{0}$$ Which of the following statements are always true?
1. A unique solution exists satisfying the initial conditions $$\displaystyle{y}{\left({0}\right)}=\pi,\ {y}'{\left({0}\right)}=\sqrt{{\pi}}$$
2. Every solution is differentiable on the interval $$\displaystyle{\left(-\infty,\infty\right)}$$
3. If $$\displaystyle{y}_{{1}}$$ and $$\displaystyle{y}_{{2}}$$ are any two linearly independent solutions, then $$\displaystyle{y}={C}_{{1}}{y}_{{1}}+{C}_{{2}}{y}_{{2}}$$ is a general solution of the equation.

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Thomas Conway
The general solution to this equation is,
$$\displaystyle{y}={C}_{{1}}{e}^{{\lambda_{{1}}{x}}}+{C}_{{2}}{e}^{{\lambda_{{2}}{x}}}$$
where, $$\displaystyle\lambda_{{1}}$$ and $$\displaystyle\lambda_{{2}}$$ are roots of the equation $$\displaystyle{a}{x}^{{2}}+{b}{x}+{c}={0}$$
1) if we know $$\displaystyle{y}{\left({0}\right)}$$ and $$\displaystyle{y}'{\left({0}\right)}$$, then we can obtain two linear equations in $$\displaystyle{C}_{{1}}$$ and $$\displaystyle{C}_{{2}}$$, 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 $$\displaystyle{\left({a}\ne{0}\right)}$$) linear equations have the uniqueness and existence property. all $$\displaystyle{y},{y}',{y}{''}$$ stay bounded in the finite part of the domain. it always has two linearly independent solutions $$\displaystyle{y}_{{1}},{y}_{{2}}$$ with $$\displaystyle{y}_{{1}}{\left({0}\right)}={1},\ {y}_{{1}}'{\left({0}\right)}={0},\ {y}_{{2}}{\left({0}\right)}={0},\ {y}_{{2}}'{\left({0}\right)}={1}$$ so that they can take care of any initial conditions. like the you have $$\displaystyle{y}{\left({0}\right)}=\pi,\ {y}'{\left({0}\right)}=\sqrt{{\pi}}$$