{:[(2sin y+1)(dy)/(dx)=4" and "],[y(0)=pi//2.]:} What is x when y=pi ? x=

Write Differential Equation: First, we have the differential equation (2sin y + 1)(dy/dx) = 4. To find x when y=pi, we need to separate variables and integrate.Separate and Integrate Variables: Separate the variables: (dy)/(2sin y + 1) = (dx)/4.Find Constant of Integration: Now, integrate both sides: ∫(dy)/(2sin y + 1) = ∫(dx)/4.Calculate Definite Integral: Integrate the left side with respect to y and the right side with respect to x: ∫(dy)/(2sin y + 1) = (1/4)x + C, where C is the constant of integration.Solve for Constant: To find C, use the initial condition y(0) = pi/2. Plug y = pi/2 and x = 0 into the integrated equation: ∫(dy)/(2sin(pi/2) + 1) from pi/2 to pi = (1/4)x + C from 0 to x.Find Particular Solution: Since sin(pi/2) = 1, the equation becomes ∫(dy)/(2*1 + 1) from pi/2 to pi = (1/4)x + C from 0 to x.Substitute Values and Solve: Simplify the integral: ∫(dy)/3 from pi/2 to pi = (1/4)x + C from 0 to x.Integrate: (1/3)y from pi/2 to pi = (1/4)x + C from 0 to x.Calculate the definite integral: (1/3)pi - (1/3)(pi/2) = (1/4)x + C.Simplify the left side: (1/3)pi - (1/6)pi = (1/4)x + C.Combine like terms: (1/6)pi = (1/4)x + C.Now, plug in x = 0 and y = pi/2 into the equation to find C: (1/6)pi = (1/4)*0 + C.Solve for C: C = (1/6)pi.Now we have the particular solution: (1/6)pi = (1/4)x + (1/6)pi.Subtract (1/6)pi from both sides to solve for x: 0 = (1/4)x.Solve for x: x = 0.

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{:[(2sin y+1)(dy)/(dx)=4" and "],[y(0)=pi//2.]:}
What is
x when
y=pi ?

x=

$\begin{array}{l} (2 \sin y+1) \frac{d y}{d x}=4 \text { and } \\ y(0)=\pi / 2 . \end{array}$ $\newline$ What is $x$ when $y=\pi$ ? $\newline$ $x=$

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Simplify variable expressions using properties

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\begin{array}{l} (2 \sin y+1) \frac{d y}{d x}=4 \text { and } \\ y(0)=\pi / 2 . \end{array}

\newline

What is

x

when

y=\pi

\newline

x=

Write Differential Equation: First, we have the differential equation 2\sin y + 1)\left(\frac{dy}{dx}\right) = 4\. To find \$x when $y=\pi$ , we need to separate variables and integrate.
Separate and Integrate Variables: Separate the variables: \frac{dy}{$2$\sin y + $1$} = \frac{dx}{$4$}\.
Find Constant of Integration: Now, integrate both sides: \(\int\frac{dy}{2\sin y + 1} = \int\frac{dx}{4}.
Calculate Definite Integral: Integrate the left side with respect to $y$ and the right side with respect to $x$ : $\int\frac{dy}{2\sin y + 1} = \frac{1}{4}x + C$ , where $C$ is the constant of integration.
Solve for Constant: To find $C$ , use the initial condition $y(0) = \frac{\pi}{2}$ . Plug $y = \frac{\pi}{2}$ and $x = 0$ into the integrated equation: $\int_{\frac{\pi}{2}}^{\pi} \frac{dy}{2\sin(\frac{\pi}{2}) + 1} = \frac{1}{4}x + C$ from $0$ to $x$ .
Find Particular Solution: Since $\sin(\frac{\pi}{2}) = 1$ , the equation becomes $\int(\frac{dy}{2\cdot 1 + 1})$ from $\frac{\pi}{2}$ to $\pi = (\frac{1}{4})x + C$ from $0$ to $x$ .
Substitute Values and Solve: Simplify the integral: $\int_{\pi/2}^{\pi} \frac{dy}{3} = \frac{1}{4}x + C$ from $0$ to $x$ .
Substitute Values and Solve: Simplify the integral: $\int_{\frac{\pi}{2}}^{\pi} \frac{dy}{3} = \frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from \frac{\pi}{2}} to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .
Substitute Values and Solve: Simplify the integral: $\int \frac{dy}{3}$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Calculate the definite integral: $\frac{\pi}{2}$ $2$ .
Substitute Values and Solve: Simplify the integral: $\int \frac{dy}{3}$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Calculate the definite integral: $\frac{\pi}{2}$ $2$ .Simplify the left side: $\frac{\pi}{2}$ $3$ .
Substitute Values and Solve: Simplify the integral: $\int \frac{dy}{3}$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Calculate the definite integral: $\frac{\pi}{2}$ $2$ .Simplify the left side: $\frac{\pi}{2}$ $3$ .Combine like terms: $\frac{\pi}{2}$ $4$ .
Substitute Values and Solve: Simplify the integral: $\int \frac{dy}{3}$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Calculate the definite integral: $\frac{\pi}{2}$ $2$ .Simplify the left side: $\frac{\pi}{2}$ $3$ .Combine like terms: $\frac{\pi}{2}$ $4$ .Now, plug in $\frac{\pi}{2}$ $5$ and $\frac{\pi}{2}$ $6$ into the equation to find $\frac{\pi}{2}$ $7$ : $\frac{\pi}{2}$ $8$ .
Substitute Values and Solve: Simplify the integral: $\int \frac{dy}{3}$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Calculate the definite integral: $\frac{\pi}{2}$ $2$ .Simplify the left side: $\frac{\pi}{2}$ $3$ .Combine like terms: $\frac{\pi}{2}$ $4$ .Now, plug in $\frac{\pi}{2}$ $5$ and $\frac{\pi}{2}$ $6$ into the equation to find $\frac{\pi}{2}$ $7$ : $\frac{\pi}{2}$ $8$ .Solve for $\frac{\pi}{2}$ $7$ : $\pi$ $0$ .
Substitute Values and Solve: Simplify the integral: $\int \frac{dy}{3}$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Calculate the definite integral: $\frac{\pi}{2}$ $2$ .Simplify the left side: $\frac{\pi}{2}$ $3$ .Combine like terms: $\frac{\pi}{2}$ $4$ .Now, plug in $\frac{\pi}{2}$ $5$ and $\frac{\pi}{2}$ $6$ into the equation to find $\frac{\pi}{2}$ $7$ : $\frac{\pi}{2}$ $8$ .Solve for $\frac{\pi}{2}$ $7$ : $\pi$ $0$ .Now we have the particular solution: $\pi$ $1$ .
Substitute Values and Solve: Simplify the integral: $\int \frac{dy}{3}$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from $\frac{\pi}{2}$ to $\pi$ = $\frac{1}{4}x + C$ from $0$ to $x$ .Calculate the definite integral: $\frac{\pi}{2}$ $2$ .Simplify the left side: $\frac{\pi}{2}$ $3$ .Combine like terms: $\frac{\pi}{2}$ $4$ .Now, plug in $\frac{\pi}{2}$ $5$ and $\frac{\pi}{2}$ $6$ into the equation to find $\frac{\pi}{2}$ $7$ .Solve for $\frac{\pi}{2}$ $8$ .Now we have the particular solution: $\frac{\pi}{2}$ $9$ .Subtract $\pi$ $0$ from both sides to solve for $\pi$ $1$ .
Substitute Values and Solve: Simplify the integral: $\int \frac{dy}{3}$ from $\pi/2$ to $\pi = \frac{1}{4}x + C$ from $0$ to $x$ .Integrate: $\frac{1}{3}y$ from $\pi/2$ to $\pi = \frac{1}{4}x + C$ from $0$ to $x$ .Calculate the definite integral: $\pi/2$ $0$ .Simplify the left side: $\pi/2$ $1$ .Combine like terms: $\pi/2$ $2$ .Now, plug in $\pi/2$ $3$ and $\pi/2$ $4$ into the equation to find $\pi/2$ $5$ : $\pi/2$ $6$ .Solve for $\pi/2$ $5$ : $\pi/2$ $8$ .Now we have the particular solution: $\pi/2$ $9$ .Subtract $\pi = \frac{1}{4}x + C$ $0$ from both sides to solve for $x$ : $\pi = \frac{1}{4}x + C$ $2$ .Solve for $x$ : $\pi/2$ $3$ .

$\begin{array}{l} (2 \sin y+1) \frac{d y}{d x}=4 \text { and } \\ y(0)=\pi / 2 . \end{array}$ $\newline$ What is $x$ when $y=\pi$ ? $\newline$ $x=$

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$\begin{array}{l} (2 \sin y+1) \frac{d y}{d x}=4 \text { and } \\ y(0)=\pi / 2 . \end{array}$ $\newline$ What is $x$ when $y=\pi$ ? $\newline$ $x=$