# Extraordinary Differential Equations #6

Calculus Level pending

The two functions $$y=y(x)$$ and $$w=w(x)$$ are such that the 2 differential equations $ay''+by'+cy=x^2$ and $w'''+dw''+ew'+fw=x^2$ share the same particular solution $y_p(x)=w_p(x)=-\frac{x^2}{6}+\frac{5x}{18}-\frac{37}{108}$ where $$a, b, c, d, e$$ and $$f$$ are integer constants to be determined.

The general solutions for both differential equations would be in the form $y(x)=c_1 e^{Jx}+c_2 e^{Kx}-\frac{x^2}{6}+\frac{5x}{18}-\frac{37}{108}$ and $w(x)=c_3 e^{Lx}+c_4 e^{Mx}+c_5 e^{Nx}-\frac{x^2}{6}+\frac{5x}{18}-\frac{37}{108}$ where $$c_1$$ through $$c_5$$ are arbitrary constants of integration, and $$J, K, L, M$$ and $$N$$ are real numbers to be determined.

Determine the sum $$\frac{1}{J+K+L+M+N}$$.

(This problem is part of the set Extraordinary Differential Equations.)

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