How can I solve this system of equation?

Question

I ham given the following problem to solve:

1.9. The program should take three numbers: a; b; c and find the roots of the quadratic equation in the form:

If the value of the determinant of the quadratic equation is negative (i.e. ∆ <0), the program should write an appropriate message.

1.10. Modify the 1.9 so that the parameters A and a are functions of the parameter ϕ, e.g.

In the program, declare functions for the above equations and save the results in an array.

I have solved the 1.9 as that was easy. But I am stuck with the following:

$f(x) = f(1 + 0.25 \sin(x)) \tag 1$
$g(x) = g(1 + 0.5 |\sin(x)|) \tag 2$

My attempt (source):

Given,

$f(x) = f(1 + 0.25 \sin(x))$

Now,

$f(1 + 0.25 \sin(x)) = \frac{a}{16} \sin^2{x} +\left (\frac{a}{2} + \frac{b}{4}\right) \sin{x} + (a+b+c)$

Let, $\sin{x} = y$

So, the above equation becomes: $y = \frac{-(2a+b) \pm \sqrt{b^2 - 4ac}} {2a}$

Hence,

$\begin{align} \sin{x} &= \dfrac{-(2a+b) \pm \sqrt{b^2 - 4ac}}{2a} \\ x &= \sin^{-1}{\left (\dfrac{-(2a+b) \pm \sqrt{b^2 - 4ac}}{2a} \right)} \end{align}$

Are this procedure and solution correct?

@ms._VerkhovtsevaKatya, no, I followed the procedure used in the source. — user366312, Apr 07 '20 at 21:49
In its current form, it is not clear what do mean for "solve". Are you trying to 1) find a function $f(x)$ that satisfies the functional equation $f(x) = f(1 + \frac14\sin x)$ for all $x$ or 2) given a specific $f(x)$ (some quadratic polynomial?), find a $x$ that satisfy the same equation. — achille hui, Apr 07 '20 at 21:59
@user366312: It is impossible "to find the value of $x$" when $f$ is not given, unless $f$ is entirely superfluous. (If so, why oh why is it included in the question?!?!) — David G. Stork, Apr 07 '20 at 22:08
Where does the line after Now come from? Why should $f$ have that form? The question you link had a condition that made it reasonable to expect its $f$ to be polynomial. Yours does not. — Ross Millikan, Apr 10 '20 at 02:03
@user366312: can you clarify what is the closed formula for $f$? — Matteo, Apr 14 '20 at 12:07
@user366312: for example $f(x)=ax^2+bx+c$ is a closed formula. — Matteo, Apr 14 '20 at 12:19
@Matteo, I asked my teacher about this. he says that it is about Maxwell's distribution. — user366312, Apr 14 '20 at 12:24

score 1 · Answer 1 · edited Apr 11 '20 at 17:34

I think that Problem 1.10 is almost as simple as Problem 1.9. The only difference is that the parameters $a$ and $b$ or $c$ (there is no parameter $A$ in Problem 1.9, so this should be a misprint) instead of constant values are functions of the parameter $\varphi$ declared in the program (so a program solving Problem 1.10 can be a small modification of the program solving Problem 1.9). A solver has the freedom to choose the functions.

The equality $A(\phi)=A(1.0+0.25\sin(\phi))$ is not an equation to solve, it is an example of a function of a parameter $\varphi$. A bit confusing here is that $A$ from the left-hand side is a name of a function and the left-hand side equals to value of $A$ of the argument $\varphi$, whereas $A$ from the right side is a constant and the right-hand side is a product of this $A$ by $1.0+0.25\sin(\phi)$. Similarly for the other equality.

How can I solve this system of equation?

1 Answers1