Solve the correct answer linear equations by considering y as a function of x, that is, \displaystyle{y}={y}{\left({x}\right)}.\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+{y}= \cos{{x}}

Solve the correct answer linear equations by considering y as a function of x, that is, \displaystyle{y}={y}{\left({x}\right)}.\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}+{y}= \cos{{x}}

Question
Solve the correct answer linear equations by considering y as a function of x, that is, \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}={\left\lbrace{y}\right\rbrace}{\left\lbrace{\left({\left\lbrace{x}\right\rbrace}\right)}\right\rbrace}.{\frac{{{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{y}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}}}{{{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}}}}+{\left\lbrace{y}\right\rbrace}={\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\)

Answers (1)

2021-03-08
Variation of parameters
First, solve the linear homogeneous equation by separating variables.
Rearranging terms in the equation gives
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\frac{{{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{y}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}}}{{{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}}}}=-{\left\lbrace{y}\right\rbrace}\Leftrightarrow{\frac{{{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{y}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}}}{{{\left\lbrace{y}\right\rbrace}}}}=-{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}\)
Now, the variables are separated, x appears only on the right side, and y only on the left.
Integrate the left side in relation to y, and the right side in relation to x
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le\int{\frac{{{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{y}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}}}{{{\left\lbrace{y}\right\rbrace}}}}=-\int{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace},\)
which is
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{\ln}\right\rbrace}{\left\lbrace\le{f}{t}{\left|{\left\lbrace{y}\right\rbrace}{r}{i}{g}{h}{t}\right|}\right\rbrace}=-{\left\lbrace{x}\right\rbrace}+{\left\lbrace{c}\right\rbrace}.\)
By taking exponents, we obtain
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{\ln}\right\rbrace}{\left\lbrace\le{f}{t}{\left|{\left\lbrace{y}\right\rbrace}{r}{i}{g}{h}{t}\right|}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}+{\left\lbrace{c}\right\rbrace}\right\rbrace}}}={\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\cdot{\left\lbrace{e}\right\rbrace}^{{{c}}}.\)
Hence,we obtain
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}={\left\lbrace{C}\right\rbrace}{\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}},\)
where \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{C}\right\rbrace}\:=\pm{\left\lbrace{e}\right\rbrace}^{{{c}}}{\left\lbrace\quad\text{and}\quad\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}={\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\) is the complementary solution.
Next, we need to find the particular solution \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{p}\right\rbrace}}}\).
Therefore, we consider \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{u}\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}\) and try to find u, a function of x, that will make this work.
Let’s assume that \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{u}\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}\) is a solution of the given equation. Hence, it satisfies the given equation. Substituting \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{u}\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}\) and its derivative in the equation gives
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{\left({\left\lbrace{u}\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}\right)}\right\rbrace}'+{\left\lbrace{u}\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}={\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{u}\right\rbrace}'{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}+{\left\lbrace{u}\right\rbrace}{\left\lbrace{y}\right\rbrace}'_{{{\left\lbrace{c}\right\rbrace}}}+{\left\lbrace{u}\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}={\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le\underbrace{{{\left\lbrace{\left\lbrace{u}\right\rbrace}'{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}+{\left\lbrace{u}\right\rbrace}{\left\lbrace{\left({\left\lbrace{y}\right\rbrace}'{\left\lbrace{c}\right\rbrace}+{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{e}\right\rbrace}}}\right)}\right\rbrace}={\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right\rbrace}}}_{{{\left\lbrace={\left\lbrace{0}\right\rbrace}\ \text{since}\ {\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}\text{is a solution}\right\rbrace}}}\)
Therefore, \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{u}\right\rbrace}'{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}={\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}{\left\lbrace{x}\right\rbrace}\Rightarrow{\left\lbrace{u}\right\rbrace}'={\frac{{{\left\lbrace{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}}}}\)
which gives
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{u}\right\rbrace}=\int{\frac{{{\left\lbrace{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}}}}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}\)
Now, we can find the function u:
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{u}\right\rbrace}=\int{\frac{{{\left\lbrace{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right\rbrace}}}{{{\left\lbrace{\left\lbrace{e}\right\rbrace}^{{-{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right\rbrace}}}}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}=\overbrace{{{\left\lbrace\int{\left\lbrace{e}\right\rbrace}^{{{x}}}{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right\rbrace}}}^{{{\left\lbrace\text{Integration by parts}\int{\left\lbrace{u}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace{v}\right\rbrace}={\left\lbrace{u}\right\rbrace}{\left\lbrace{v}\right\rbrace}-\int{\left\lbrace{v}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace{u}\right\rbrace}\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace\le{f}{t}{\left|{b}{e}{g}\in{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}{\left\lbrace{u}\right\rbrace}={\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\Rightarrow{\left\lbrace{d}\right\rbrace}{\left\lbrace{u}\right\rbrace}=-{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}\backslash{\left\lbrace{d}\right\rbrace}{\left\lbrace{v}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{x}}}\Rightarrow{\left\lbrace{v}\right\rbrace}=\int{\left\lbrace{d}\right\rbrace}{\left\lbrace{v}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{x}}}{e}{n}{d}{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right|}\right\rbrace}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace{e}\right\rbrace}^{{{x}}}\cdot{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}-\int{\left\lbrace{e}\right\rbrace}^{{{x}}}{\left\lbrace{\left(-{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}\right)}\right\rbrace}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{e}\right\rbrace}^{{{x}}}\cdot{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+\overbrace{{{\left\lbrace\int{\left\lbrace{e}\right\rbrace}^{{{x}}}{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right\rbrace}}}_{{{\left\lbrace\text{Integration by parts}\int{\left\lbrace{u}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace{v}\right\rbrace}={\left\lbrace{u}\right\rbrace}{\left\lbrace{v}\right\rbrace}-\int{\left\lbrace{v}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace{u}\right\rbrace}\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace\le{f}{t}{\left|{b}{e}{g}\in{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}{\left\lbrace{u}\right\rbrace}={\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\Rightarrow{\left\lbrace{d}\right\rbrace}{\left\lbrace{u}\right\rbrace}={\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}\backslash{\left\lbrace{d}\right\rbrace}{\left\lbrace{v}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{x}}}\Rightarrow{\left\lbrace{v}\right\rbrace}=\int{\left\lbrace{d}\right\rbrace}{\left\lbrace{v}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{x}}}{e}{n}{d}{\left\lbrace{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right|}\right\rbrace}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace{e}\right\rbrace}^{{{x}}}\cdot{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\left\lbrace{e}\right\rbrace}^{{{x}}}{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}-\underbrace{{{\left\lbrace\int{\left\lbrace{e}\right\rbrace}^{{{x}}}{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}\right\rbrace}}}_{{{\left\lbrace\text{This is}\ {\left\lbrace{I}\right\rbrace}\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le\Rightarrow{\left\lbrace{I}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{x}}}\cdot{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\left\lbrace{e}\right\rbrace}^{{{x}}}{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}-{\left\lbrace{I}\right\rbrace}\)
Now, let`s solve for I.
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{2}\right\rbrace}{\left\lbrace{I}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{x}}}\cdot{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\left\lbrace{e}\right\rbrace}^{{{x}}}{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\left\lbrace{c}\right\rbrace}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{I}\right\rbrace}={\frac{{{1}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{e}\right\rbrace}^{{{x}}}{\left\lbrace{\left({\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right)}\right\rbrace}+{\left\lbrace{c}\right\rbrace}\)
Since we need to find only one function that will make this work, we don’t need to introduce the constant of integration c. Hence,
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{u}\right\rbrace}={\frac{{{1}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{e}\right\rbrace}^{{{x}}}{\left\lbrace{\left({\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right)}\right\rbrace}\)
Recall that \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{p}\right\rbrace}}}={\left\lbrace{u}\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}\). Therefore,
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{p}\right\rbrace}}}-{\frac{{{1}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{e}\right\rbrace}^{{{x}}}{\left\lbrace{\left({\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right)}\right\rbrace}\cdot{\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}={\frac{{{1}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{\left({\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right)}\right\rbrace}\)
The general solution is
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}={\left\lbrace{C}\right\rbrace}{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{c}\right\rbrace}}}+{\left\lbrace{y}\right\rbrace}_{{{\left\lbrace{p}\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace{C}\right\rbrace}{\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\frac{{{1}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{e}\right\rbrace}^{{{x}}}\)
Integrating Factor technique
This equation is linear with \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{P}\right\rbrace}{\left\lbrace{\left({\left\lbrace{x}\right\rbrace}\right)}\right\rbrace}={\left\lbrace{1}\right\rbrace}{\left\lbrace\quad\text{and}\quad\right\rbrace}{\left\lbrace{Q}\right\rbrace}{\left\lbrace{\left({\left\lbrace{x}\right\rbrace}\right)}\right\rbrace}={\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}.\)
Hence,
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{h}\right\rbrace}=\int{\left\lbrace{P}\right\rbrace}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}=\int{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}={\left\lbrace{x}\right\rbrace}\)
So, an integrating factor is \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{e}\right\rbrace}^{{{h}}}={\left\lbrace{e}\right\rbrace}^{{{x}}}\)
and the general solution is
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}{\left\lbrace{\left({\left\lbrace{x}\right\rbrace}\right)}\right\rbrace}={\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{h}\right\rbrace}\right\rbrace}}}{\left\lbrace{\left({\left\lbrace{c}\right\rbrace}+\int{\left\lbrace{Q}\right\rbrace}{\left\lbrace{e}\right\rbrace}^{{{h}}}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}\right)}\right\rbrace}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}{\left\lbrace{\left({\left\lbrace{c}\right\rbrace}+\int{\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\cdot{\left\lbrace{e}\right\rbrace}^{{{x}}}{\left\lbrace\le{f}{t}.{\left\lbrace{d}\right\rbrace}{\left\lbrace{x}\right\rbrace}{r}{i}{g}{h}{t}.\right\rbrace}\right)}\right\rbrace}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}{\left\lbrace{\left({\left\lbrace{c}\right\rbrace}+{\frac{{{1}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{e}\right\rbrace}^{{{x}}}{\left\lbrace{\left({\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right)}\right\rbrace}\right)}\right\rbrace}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace{c}\right\rbrace}{\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\frac{{{1}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{\left({\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right)}\right\rbrace}\)
We get the finally answer is
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}={\left\lbrace{c}\right\rbrace}{\left\lbrace{e}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\frac{{{1}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{\left({\cos{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}+{\sin{{\left\lbrace{\left\lbrace{x}\right\rbrace}\right\rbrace}}}\right)}\right\rbrace}\)
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