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Is there any analytic solution for $f(x)=0$?

Where $f(x)=A\cos{x}+ B\sin{x}+C\sin{x}\cos{x}+D$

and

$x \in \left[ 0,\frac{\pi}{2} \right]$

and

$A,B,C,D \in \mathbb{R}$

P.S.: (For those ones who may ask about my intention for asking the question) For my research purpose, I have derived a closed-form equation for a harmonic current in a specific power electronics circuit. The derivative of the formula is $f(x)$ which is purposed for finding local maximums.

Hamid
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  • Differentiate twice and then add side by side. You will get rid of first two elements. This should give $C=0$ and $D=0$. From this $A=0$ and $B=0$. – Salcio Jan 13 '22 at 02:59
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    The Weierstrass substitution will turn it into a quartic equation, which has closed form solutions but they are not pretty. – dxiv Jan 13 '22 at 03:01
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    You get the same approach as the comment of @dxiv by noting that $$\cos(x)[A + C\sin(x)] = -D - B\sin(x).$$ Isolating $\cos(x)$ and then squaring both sides leads to $$1 - \sin^2(x) = \cos^2(x) = \left[\frac{-D - B\sin(x)}{A + C\sin(x)}\right]^2.$$ This leads to a quartic equation in $\sin(x)$, where each of the $4$ roots of this quartic are merely candidate solutions. The RHS fraction above assumes that $A + C\sin(x) \neq 0.$ – user2661923 Jan 13 '22 at 03:17
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    @Salcio Differentiating an equation to give you another equation does not, in general, help you solve the first equation. Except in special circumstances, such as when the first equation is an identity. See, for example, my answer to this question: https://math.stackexchange.com/questions/1952774/it-is-possible-to-apply-a-derivative-to-both-sides-of-a-given-equation-and-maint – Deepak Jan 13 '22 at 03:38
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    @Deepak but we are not talking about a general equation but the specific one. Here, second derivative of $cox$ is $-cosx$ and the same for $sinx$n hence doing this and adding side by side removes two variables $A$ and $B$. – Salcio Jan 13 '22 at 13:00
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    @Salcio: The objection is that here $x$ satisfies the given equation, not the one you would get by differentiating (twice!). One does not solve a quadratic polynomial equation by differentiating both sides. – hardmath Jan 17 '22 at 13:58
  • @dxiv! Thanks a lot for the helpful advice. – Hamid Jan 18 '22 at 04:19
  • @user2661923! Thanks a lot for the helpful advice. – Hamid Jan 18 '22 at 04:20
  • @Salcio Thanks a lot for your advice. – Hamid Jan 18 '22 at 04:21
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    @hardmath! I add some comments to my question, I hope it is helpful. – Hamid Jan 18 '22 at 04:22
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    Yes, thanks for that. Applied problems often have a nice solution. Even though the quartic polynomial approaches outlined above are analytic, I'd expect the polynomial solution you want is most efficiently found by a numerical root finder. Note $\sin x \in [0,1]$. – hardmath Jan 18 '22 at 05:47

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