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

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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

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

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