Determine the order of the given differential equation and state whether the equation is linear
or nonlinear.

(a) The order of this differential equation is
(b) The equation is Choose

Ask by Evans Park. in the United States

Mar 23,2025

Question

Determine the order of the given differential equation and state whether the equation is linear
or nonlinear.

(a) The order of this differential equation is
(b) The equation is Choose

Ask by Evans Park. in the United States

Mar 23,2025

UpStudy AI · Accepted Answer

To analyze the given differential equation

$$
(1-x) y^{\prime \prime}-4 x y^{\prime}+2 y=\cos (x),
$$

we will determine the order and whether it is linear or nonlinear.

### (a) Order of the Differential Equation
The order of a differential equation is defined as the highest derivative present in the equation. In this case, we have:

- $$ y^{\prime \prime} $$ (the second derivative of $$ y $$)
- $$ y^{\prime} $$ (the first derivative of $$ y $$)
- $$ y $$ (the function itself)

The highest derivative is $$ y^{\prime \prime} $$, which is the second derivative. Therefore, the order of this differential equation is **2**.

### (b) Linearity of the Equation
A differential equation is considered linear if it can be expressed in the form:

$$
a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \ldots + a_1(x) y' + a_0(x) y = g(x),
$$

where $$ a_n(x), a_{n-1}(x), \ldots, a_0(x) $$ are functions of $$ x $$ only, and $$ g(x) $$ is also a function of $$ x $$. Additionally, the dependent variable $$ y $$ and its derivatives must appear to the first power and must not be multiplied together.

In the given equation:

$$
(1-x) y^{\prime \prime}-4 x y^{\prime}+2 y=\cos (x),
$$

- The coefficients $$ (1-x) $$, $$ -4x $$, and $$ 2 $$ are functions of $$ x $$.
- The dependent variable $$ y $$ and its derivatives $$ y^{\prime} $$ and $$ y^{\prime \prime} $$ appear to the first power and are not multiplied together.

Thus, the equation is **linear**.

### Summary
(a) The order of this differential equation is **2**.  
(b) The equation is **linear**.
(a) The order of the differential equation is **2**. (b) The equation is **linear**.

Determine the order of the given differential equation and state whether the equation is linear
or nonlinear.

(a) The order of this differential equation is
(b) The equation is Choose

Upstudy AI Solution

Answer

Solution

(a) Order of the Differential Equation

(b) Linearity of the Equation

Summary

The Deep Dive

Related Questions

Latest Calculus Questions

Determine the order of the given differential equation and state whether the equation is linear or nonlinear. (a) The order of this differential equation is (b) The equation is Choose

Upstudy AI Solution

Answer

Solution

(a) Order of the Differential Equation

(b) Linearity of the Equation

Summary

The Deep Dive

Related Questions

Latest Calculus Questions

Determine the order of the given differential equation and state whether the equation is linear
or nonlinear.

(a) The order of this differential equation is
(b) The equation is Choose