Certainly! Let's classify the given differential equation step-by-step:
[tex]\[
\frac{d^4 y}{d x^4} + \frac{d^3 y}{d x^3} + \frac{d^y}{d x^2} + \frac{d y}{d x} + y = 1
\][/tex]
(a) Type:
To determine the type of the differential equation, we need to see what kind of derivatives are involved. Since the equation involves ordinary derivatives (derivatives with respect to a single variable [tex]\(x\)[/tex]), it is an Ordinary Differential Equation (ODE).
Therefore, the type is:
- Ordinary Differential Equation
(b) Order:
The order of a differential equation is determined by the highest derivative present in the equation. In this case, we have derivatives of [tex]\(y\)[/tex] up to the fourth derivative, i.e., [tex]\(\frac{d^4 y}{d x^4}\)[/tex].
Thus, the order of this differential equation is:
- 4
(c) Linearity:
A differential equation is linear if the dependent variable [tex]\(y\)[/tex] and all its derivatives appear to the power of one and are not multiplied together. Here, [tex]\(y\)[/tex] and its derivatives [tex]\(\frac{d y}{d x}\)[/tex], [tex]\(\frac{d^2 y}{d x^2}\)[/tex], [tex]\(\frac{d^3 y}{d x^3}\)[/tex], and [tex]\(\frac{d^4 y}{d x^4}\)[/tex] all appear linearly (to the first power and are not multiplied by each other).
Hence, this differential equation is:
- Linear
So, putting it all together:
- (a) Type: Ordinary Differential Equation
- (b) Order: 4
- (c) Linearity: Linear
