Differential Equation and Applications

(x² + y²)dx + xy dy = 0, x dx + y dy = 0, y dx - x dy = 0

Step 1: Check homogeneity by verifying if f(tx,ty) = t^n f(x,y) for both functions. Option 1: f₁ = x² + y², f₂ = xy. Both have degree 2. ✓ Option 2: f₁ = x, f₂ = y. Both have degree 1. ✓ Option 3: f₁ = x + y (degree 1), f₂ = x (degree 1), but the equation structure makes it non-homogeneous. Option 4: f₁ = y, f₂ = -x. Both have degree 1. ✓ Option 5: f₁ = x² (degree 2), f₂ = y (degree 1). Different degrees. ✗

k = (ln 2)/5

Step 1: Population growth model is dP/dt = kP with solution P = P₀e^(kt). Step 2: Given that population doubles in 5 years: P(5) = 2P₀. Step 3: Substitute: 2P₀ = P₀e^(5k). Step 4: Simplify: 2 = e^(5k). Step 5: Take natural log: ln(2) = 5k. Step 6: Solve for k: k = ln(2)/5.

y = Cx

Step 1: Start with dy/dx = y/x. Step 2: Separate variables: dy/y = dx/x. Step 3: Integrate both sides: ∫dy/y = ∫dx/x. Step 4: ln|y| = ln|x| + ln|C|. Step 5: Using properties of logarithms: ln|y| = ln|Cx|. Step 6: Therefore: y = Cx (where C is an arbitrary constant).

The general solution contains arbitrary constants, A first-order equation can have at most one arbitrary constant, Degree is always a positive integer, The particular solution has no arbitrary constants

Statement 1: True - General solutions always contain arbitrary constants equal to the order. Statement 2: False - Order can be less than degree (e.g., (dy/dx)³ = x has order 1, degree 3). Statement 3: True - Number of constants equals order of equation. Statement 4: True - Degree is the power of highest derivative, always positive integer. Statement 5: True - Particular solutions have specific values for all constants.