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How can I prove that there exists infinite number of solutions of an ODE of order $n$:

$p_{0}\frac{d^ny}{dx^n}+p_{1}(x)\frac{d^{n-1}y}{dx^{n-1}}+...+p_{n}=0$

where $p_0, p_1,..., p_n $ all are continuous in $I$ and $p_0\ne 0 $ in $I$.

Thanks in advance!

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closed as off-topic by caverac, Decaf-Math, Adrian Keister, Theo Bendit, Cesareo Jan 31 at 17:24

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  • $\begingroup$ Consult a graduate-level textbook on differential equations. See "existence theorems" in the index. The set of solutions turns out to be an $n$-dimensional vector space, which is much more precise than merely "an infinite number". $\endgroup$ – GEdgar Jan 31 at 13:17
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    $\begingroup$ If $y=f(x)$ is a solution then $y=f(x)+c$ for any constant $c$ is also a solution because the value of each term on the left hand side is unchanged. $\endgroup$ – gandalf61 Jan 31 at 14:00
  • $\begingroup$ Or if the previous comment hints to a typing error, correct the last term in the equation which should probably be $...+p_n(x)y(x)=0$. If you use your previous question, every combination of initial values gives a unique solution, and the space of initial conditions is infinite, as it is $\Bbb R^n$. $\endgroup$ – LutzL Jan 31 at 14:21