# Prove that a nth order differential equation possesses infinite solutions [closed]

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 $$I$$.

• 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". – GEdgar Jan 31 at 13:17
• 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. – gandalf61 Jan 31 at 14:00
• 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 $\Bbb R^n$. – LutzL Jan 31 at 14:21