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Find the particular solution,
y=f(x), to the differential equation
(dy)/(dx)=(cos x)/(y) given
f((3pi)/(2))=-1

Find the particular solution, $y=f(x)$ , to the differential equation $\frac{d y}{d x}=\frac{\cos x}{y}$ given $f\left(\frac{3 \pi}{2}\right)=-1$

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Find derivatives of sine and cosine functions

Full solution

Q. Find the particular solution,

y=f(x)

, to the differential equation

\frac{d y}{d x}=\frac{\cos x}{y}

given

f\left(\frac{3 \pi}{2}\right)=-1

Separate and Integrate: Step $1$ : Separate variables and integrate. $\newline$ We start by separating the variables in the differential equation $\frac{dy}{dx} = \frac{\cos x}{y}$ . Rearrange to get $y dy = \cos x dx$ . Now, integrate both sides: $\newline$ $\int y dy = \int \cos x dx$ $\newline$ $\frac{1}{2}y^2 = \sin x + C$
Solve for y: Step $2$ : Solve for y. $\newline$ To find y, we take the square root of both sides: $\newline$ $y = \pm\sqrt{2(\sin x + C)}$
Find $C$ with Initial Condition: Step $3$ : Use the initial condition to find $C$ . We know that $f\left(\frac{3\pi}{2}\right) = -1$ . Plugging $x = \frac{3\pi}{2}$ into $y = \pm\sqrt{2(\sin x + C)}$ : $-1 = \pm\sqrt{2(\sin\left(\frac{3\pi}{2}\right) + C)}$ Since $\sin\left(\frac{3\pi}{2}\right) = -1$ , $-1 = \pm\sqrt{2(-1 + C)}$ Squaring both sides gives: $1 = 2(-1 + C)$ $1 = -2 + 2C$ $C$ $0$ $C$ $1$
Write Particular Solution: Step $4$ : Write the particular solution. $\newline$ With $C = \frac{3}{2}$ , the solution becomes: $\newline$ $y = \pm\sqrt{2(\sin x + \frac{3}{2})}$ $\newline$ Since we need the solution that passes through $(\frac{3\pi}{2}, -1)$ and we calculated earlier: $\newline$ $-1 = -\sqrt{2(\sin(\frac{3\pi}{2}) + \frac{3}{2})}$ $\newline$ The correct sign is negative, so: $\newline$ $y = -\sqrt{2(\sin x + \frac{3}{2})}$

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