Question

Which of the following equations is a linear equation of order \(3 ?\) (A) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right) \cdot(\mathrm{dy} / \mathrm{dx})+\mathrm{y}=\mathrm{x}\) (B) \(\left(\mathrm{d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)+\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)+\mathrm{y}^{2}=\mathrm{x}^{2}\) (C) \(x \cdot\left(d^{3} y / d x^{3}\right)+\left(d^{3} y / d x^{3}\right)=e^{x}\) (D) \(\left(d^{2} y / d x^{2}\right)+(d y / d x)=\log x\)

Short answer

The correct choice is option C: \( x\cdot\left(\frac{d^3y}{dx^3}\right) + \left(\frac{d^3y}{dx^3}\right) = e^x \). This equation is a linear differential equation of order 3, as it has the highest derivative term of order 3 (\(\frac{d^3y}{dx^3}\)) and no higher order terms, products, or powers involving the dependent variable (y) or its derivatives.

Step-by-step solution

Examine each option

Let's look at each equation (options A, B, C, and D) to determine whether it is a linear differential equation of order 3.

Option A

\( \left(\frac{d^3y}{dx^3}\right) + \left(\frac{d^2y}{dx^2}\right)\cdot\left(\frac{dy}{dx}\right) + y = x \) This option contains a product term: \(\left(\frac{d^2y}{dx^2}\right)\cdot\left(\frac{dy}{dx}\right)\), meaning it is not a linear equation. So option A is not the answer.

Option B

\( \left(\frac{d^3y}{dx^3}\right) + \left(\frac{d^2y}{dx^2}\right) + y^2 = x^2 \) This option has a power term: \(y^2\), meaning it is not a linear equation. Therefore, option B is not the answer.

Option C

\( x\cdot\left(\frac{d^3y}{dx^3}\right) + \left(\frac{d^3y}{dx^3}\right) = e^x \) This option has the highest derivative term of order 3 (\(\frac{d^3y}{dx^3}\)) and no higher order terms, products, or powers involving the dependent variable (y) or its derivatives. This equation is linear of order 3. Option C is the answer.

Option D (for completeness)

\(\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right) = \log x\) This option has the highest derivative term of order 2, not 3. Consequently, it is not a linear equation of order 3. The correct choice is option C.

Key concepts

Differential Equations

Differential equations are a type of equation that involves derivatives of a function. These equations are fundamental in expressing how a particular quantity changes with respect to another. For instance, they can model how temperature varies over time or the speed at which a skydiver falls due to gravity. A differential equation typically consists of a dependent variable, like temperature or position, and an independent variable, such as time, along with its derivatives.

Differential equations come in various forms and complexities, from simple separable ordinary differential equations to more complicated partial differential equations involving multiple independent variables. The beauty of differential equations lies in their ability to model and predict complex dynamic systems in the natural and engineering sciences.

Order of Differential Equation

The order of a differential equation is determined by the highest derivative that appears in the equation. To illustrate, consider the two functions: One involving the first derivative, like \(\frac{dy}{dx} = x^2\), which is a first-order differential equation; and another that involves a second derivative, such as \(\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0\), which is a second-order differential equation.

Understanding the order of a differential equation is crucial, as it affects the methods used to solve the equation as well as the nature of the solutions. Typically, a higher-order differential equation can be associated with more complex systems and often requires more boundary conditions to find a unique solution. In the context of our original exercise, a third-order differential equation implies the equation deals with the third derivative of the dependent variable.

Linearity in Differential Equations

Linearity in differential equations means that the dependent variable and its derivatives appear to the first power and are not multiplied or composed with each other. A linear differential equation will have the general form \(a_n(x)\frac{d^ny}{dx^n} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x)\), where \(a_0(x), a_1(x), \) and so on, up to \(a_n(x)\), are functions of the independent variable \(x\) alone, and \(g(x)\) is another function known as the inhomogeneous term.

In the example of option C from the exercise, \(x\left(\frac{d^3y}{dx^3}\right) + \left(\frac{d^3y}{dx^3}\right) = e^x\) adheres to the definition of linearity for third-order differential equations. Notably, it doesn't include any terms where the dependent variable or its derivatives are raised to a power other than one or multiplied together, qualifying it as a linear differential equation of order 3.