Differential Equations and Modeling

y = e^(-x), y = 2e^(-x), y = -3e^(-x), y = ce^(-x)

Step 1: Solve dy/dx = -y by separation: dy/y = -dx. Step 2: Integrate: ln|y| = -x + C. Step 3: y = Ae^(-x) where A is any constant. All options with form Ae^(-x) are correct, while e^x gives dy/dx = e^x ≠ -y.

k = (ln 2)/2

Step 1: Solve dP/dt = kP: P = P₀e^(kt). Step 2: Using P(0) = 100: P = 100e^(kt). Step 3: Using P(2) = 200: 200 = 100e^(2k). Step 4: Solve: 2 = e^(2k), so 2k = ln 2, therefore k = (ln 2)/2.

Order = 2, Degree = 1

Step 1: Identify highest derivative: y'' = d²y/dx² (second order). Step 2: Check polynomial form: equation is polynomial in derivatives. Step 3: Find power of highest derivative: (y'')¹, so degree = 1. Therefore: Order = 2, Degree = 1.

y = cx

Step 1: Separate variables: dy/y = dx/x. Step 2: Integrate both sides: ∫dy/y = ∫dx/x. Step 3: ln|y| = ln|x| + ln|c|. Step 4: ln|y| = ln|cx|, so y = cx.