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Let $R_2$ be a second-order linear differential equation with non-constant coefficients, and R_2[y] = 0. Then P is a polynomial?
Let $R_2$ be a second-order linear differential equation with non-constant coefficients, and $R_2[y] = 0$. Then P is a polynomial?
I think this should be correct if it’s true that any polynomial has a closed-form solution by its defintion. However, I’m not sure how to approach this problem.
Thank you for any help.
A:
The first thing to do is to show that $p(y)=a_0+a_1y+a_2y^2$ is a solution to your differential equation. The second thing to do is to prove that your differential equation actually has a closed-form solution, i.e. that your equation admits a general solution of the form $y=y_1(x)+y_2(x)$. Let’s do that now:
Since $R_2[y] = 0$ then $R_2(y)=a_3y+a_4y^2+a_5y^3$, and therefore$$R_2(p)=a_3p+2a_4y_2+a_5y_2^2=a_3+2a_4y_1+a_5y_1^2=\dots$$In general, this means that$$y_1=-\frac{1}{a_3}R_2′(p)\implies y_1’=-R_2′(p)^{ -1}R_2′(p)=-a_3p’+2a_4y_1+a_5y_1^2=\dots\implies y_1=\frac{1}{a_3}\left(a_4e^x+a_5e^{2x}\right)$$Now we have that $y_1$ is a solution to
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