Differential Equations

dy/dx = xy, dy/dx = y/x, dy/dx = sin(x)cos(y), dy/dx = x²y³

Variable separable equations have the form dy/dx = f(x)g(y). Options 1, 3, 4, and 5 can be written as products of functions of x and y separately: xy = x·y, y/x = (1/x)·y, sin(x)cos(y) = sin(x)·cos(y), x²y³ = x²·y³. Option 2 (x + y) cannot be separated as a product.

1

This is a first-order differential equation. The general solution of an nth order differential equation contains n arbitrary constants. Since this is first order, the general solution y = x³ + C contains 1 arbitrary constant C.

y = Cx

Step 1: x dy - y dx = 0 → x dy = y dx. Step 2: Separate variables: dy/y = dx/x. Step 3: Integrate both sides: ∫dy/y = ∫dx/x → ln|y| = ln|x| + ln|C| = ln|Cx|. Step 4: Therefore, |y| = |Cx| → y = Cx.

dy/dx + y = x, (dy/dx)² = 4y, dy/dx = sin(x)

First-order differential equations contain only first derivatives (dy/dx) as the highest order derivative. Options 1, 3, and 4 contain only dy/dx terms. Options 2 and 5 contain second and third derivatives respectively, making them higher-order equations.