Find the number of solutions for the equation $(F(x))^2=frac{9x^4}{G(x)}$

Question

Let $f_1(x)$ and $f_2(x)$ be twice differentiable functions, where $F(x)=f_1(x)+f_2(x)$ and $G(x)=f_1(x)-f_2(x)$, for all $x in mathbb{R}$, $f_1(0)=2$ and $f_2(0)=1$. If $f_1'(x)=f_2(x)$ and $f_2'(x)=f_1(x)$, for all $x in mathbb{R}$, then the number of solutions of the equation $(F(x))^2=frac{9x^4}{G(x)}$ is
I have no idea how to start. Please help.

Peanut · Accepted Answer

$f_1'' = (f_1')' = (f_2)' = f_1$, which gives $f_1 = c_1e^x+c_2e^{-x}$. From here $f_1(0) = 2$ and $f_1'(0) = f_2(0) = 1$ give $c_1 = 3/2$ and $c_2 = 1/2$. Can you go on?

Find the number of solutions for the equation $(F(x))^2=frac{9x^4}{G(x)}$

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