Step-by-step explanation:
From the information given:
y' = y - x²
y' - y = -x²
This is a first-order linear differential equation of the form y' + p(x)y = q(x)
Thus, y is dependent and x is said to be the independent variable.
xy' = 2y
This is also a first-order linear differential equation with y as the dependent variable and x as the independent variable.
x'' + 5x = e⁻ˣ
This is a second order non-linear differentiall equation.
Here, x is the dependent variable, and since the independent variable is not given, we will take t as the independent variable.
Equation independent dependent order linear/non linear
variable variable
y' = y - x² x y 1 linear
xy' = 2y x y 1 linear
x'' + 5x - e⁻ˣ t x 2 non-linear
Why this is true for (c) i.e. being non-linear is that; in the above two first-order linear differential equations, we will notice that the dependent variables and their derivatives are not multiplied together, but in the third equation e⁻ˣ which is not the first degree, the case is different as