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

Mathematics Asked by Identicon on November 11, 2020

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

$$f_1'' = (f_1')' = (f_2)' = f_1$$, 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?

Correct answer by Peanut on November 11, 2020

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