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

Related Questions

Proving that the integral closure of $R$ in $S$ is a subring of $S.$

2  Asked on February 22, 2021

Right versus left derivative by increasing one-sided derivatives

2  Asked on February 22, 2021 by wyatt

Properties of a certain matrix in a certain field

1  Asked on February 22, 2021

How do I estimate $int_{2}^{infty} frac{|log log t|}{t, (log t)^2} , dt$

2  Asked on February 22, 2021 by user330477

Getting the kinematic equation $v^2=v_0^2+2a(x−x_0)$

4  Asked on February 22, 2021 by slifjo

Suppose $P(X > x) le 100 (e^{-ax}+e^{-bx})$. Is there a clever way to bound $mathbb E[X]$?

1  Asked on February 22, 2021

Locus of a complex number $z$ when locus of $z^2$ is known

5  Asked on February 21, 2021 by arshiya

Finding all possible Jordan Normal Forms

1  Asked on February 21, 2021 by michael-maier

Finding $lim_{n to infty} int_{2}^{infty} frac{nsinleft(frac{x-2}{n}right)}{(x-2)+(1+(x-2)^2)} dx$

1  Asked on February 21, 2021 by user160

A matrix cannot be expressed as a sum $B+C$ with $B^2=B$ and $C^3=0$

1  Asked on February 21, 2021 by mashed-potato

Trigonometric system with two unknowns

0  Asked on February 21, 2021 by melisa-karimi

Noncompact (Lie) group has no faithful, finite dimensional, and unitary representations?

2  Asked on February 21, 2021 by annie-marie-heart

How to denote a one to one mapping between $A = { B,C,D}$ and $B = { {X_1},{X_2},{X_3}}$?

0  Asked on February 21, 2021 by tuong-nguyen-minh

Evaluate $int_{0}^infty frac{sin(x)}{sqrt{x^2+1}} dx$

1  Asked on February 20, 2021 by stanislas-castellana

Dimension of the intersection of $k > n$ hyperplanes

1  Asked on February 20, 2021 by canine360

Existence of a monotonic-decreasing function with monotonically increasing log derivative

0  Asked on February 20, 2021 by sam-blitz

For a square matrix $A$, if $AB = 0$ for some nonzero $B$ then $CA = 0$ for a nonzero matrix $C$.

3  Asked on February 20, 2021

Regular differentials on a singular curve.

0  Asked on February 20, 2021 by red_trumpet

For which $N$ is it possible to arrange all whole numbers from $1$ to $N$ in such a way that every adjacent pair sums up to a Fibonacci number?

1  Asked on February 20, 2021 by rubenscube

Hint Needed for a Decreasing Sequence of Limit Points

1  Asked on February 19, 2021