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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