Solve the following system of equations for a,b and c

Question

$ax+bx^{-1}+cx^3 = 0 \ (1) $

$a - bx^{-2} + 3cx^2 = 0 \ (2)$

$bx^{-3} + 6cx = x\cos x \ (3)$

If the coefficients were constants I would have used matrix row operations however this is not the case.

Answer 1

Consider (1) + (2)$x$: $$ax + bx^{-1} + cx^3 + ax - bx^{-1} + 3cx^3 = 0$$ $$2ax + 4cx^3 = 0$$ $$ax + 2cx^3 = 0$$ $$a + 2cx^2 = 0,~\text{or}~x = 0$$

Consider (2) + (3)$x$: $$a - bx^{-2} + 3cx^2 + bx^{-2} + 6cx^2 = x^2\cos x$$ $$a + 9cx^2 = x^2\cos x$$

Assuming $x \not = 0$ and using the above two results,

$$7cx^2 = x^2\cos x$$ $$7c = \cos x$$

With $c$ in terms of $x$, it is then trivial to find $a, b$ in terms of $x$ using

$$a = -2cx^2$$ $$b = x^2(a + 3cx^2)$$