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I came across the following problem that says:

The number of non-trivial polynomial solutions of the differential equation $x^3y'(x)=y(x^2)$ is which of the following?
$(1)0\space (2)1 \space (3)3 (4)\infty.$

Can someone point me in the right direction? Thanks in advance for your time.

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Does the $y(x^2)$ on the right side mean $y \cdot x^2$, or on the other hand that the function $y$ is evaluated at input $x^2$? – coffeemath Mar 7 '13 at 17:44
It is the function $y$ which is evaluated at input $x^2.$ – user52976 Mar 7 '13 at 17:46
The only polynomial that could fit such a description is quadratic. So try $y=ax^2+bx+c$ and see the resulting system of equations by setting the coefficients of both sides equal. – Maesumi Mar 7 '13 at 17:51
up vote 3 down vote accepted

For a polynomial of order $n$, this reads: $$3+(n-1)=2n \to n=2$$ Write this out: $$2ax^4+bx^3=ax^4+bx^2+c \to ax^4+bx^2(x-1)-c=0$$ So, the answer is zero, since no such non-trivial polynomials exist (non-trivial forth degree polynomials have at most 4 real roots).

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Well, if you proceed using the ansatz (thanks @user33640 for the "correction") $$ y(x) = \sum_{n=0}^N a_n x^n $$ and substitute it directly in the equation, you end up with $$ x^3 \sum_{n=1}^N n a_n x^{n-1} = \sum_{n=0}^N a_n x^{2n}. $$ This means that $$ \sum_{n=1}^N n a_n x^{n+2} - \sum_{n=0}^N a_n x^{2n} = 0 $$ which tells you that all $a_n = 0$ for $n=2m+1$. This in turn help us to propose an improved ansatz $$ y(x) = \sum_{n=0}^N b_{2n} x^{2n}. $$ Again, substituting in the ode, we have $$ \sum_{n = 1}^N 2n b_{2n} x^{2n+2} = \sum_{n=0}^N b_{2n} x^{4n} $$ Now, the only way to this equation to be satisfied, is if $$ 2N + 2 = 4N \quad \Longrightarrow \quad N=1 $$ Then $y(x) = b_0 + b_2 x^2$ implies that $$ 2 b_2 x^4 = b_0 + b_2 x^4 $$ which means $b_0 = b_2 = 0$ hence no non-trivial solution exist.

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what is "anzat?" – user52976 Mar 7 '13 at 18:13
@user33640: it should be ansatz a trial solution which is to be verified. – Ross Millikan Mar 7 '13 at 20:56
@user33640 According to Wookipedia, "anzats" were a dangerous and mysterious Force-sensitive near-Human species with two tentacle-like proboscises that curled out and extended from their cheeks, with which the Anzati were able to feed upon the brains of their prey. An ansatz is a proposed form for the solution of an equation :) – Pragabhava Mar 7 '13 at 20:57

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