Question

Classify the following ordinary differential equations: (a) \(\quad y^{\prime \prime}+3 y^{\prime}+4 y=0\). (b) \(y^{\prime \prime \prime}+2 y^{\prime \prime}+5 y y^{\prime}=x\). (c) \(y^{i v}+y=\sin x\). (d) \(y^{\prime \prime}+(\cos x) y^{\prime}+x^{2} y=0\). (e) \(\left(y^{\prime \prime}\right)^{2}+y^{\prime}=e^{x}\). (f) \(y y^{\prime \prime}+2 x y^{\prime}+y=0\). (g) \(y^{\prime \prime \prime}+\left(y^{\prime \prime}\right)^{2} y^{\prime}+y=5\).

Short answer

(a) The given ODE is \(y^{\prime \prime} + 3y^{\prime} + 4y = 0\). This is classified as a second-order linear ODE. (b) The given ODE is \(y^{\prime \prime \prime} + 2y^{\prime \prime} + 5yy^{\prime} = x\). This is classified as a third-order nonlinear ODE. (c) The given ODE is \(y^{iv}+y=\sin x\). This is classified as a fourth-order linear ODE. (d) The given ODE is \(y^{\prime \prime}+(\cos x) y^{\prime}+x^{2} y=0\). This is classified as a second-order linear inhomogeneous ODE. (e) The given ODE is \(\left(y^{\prime \prime}\right)^{2}+y^{\prime}=e^{x}\). This is classified as a nonlinear second-order ODE. (f) The given ODE is \(yy^{\prime \prime} + 2xy^{\prime} + y = 0\). This is classified as a second-order nonlinear ODE. (g) The given ODE is $y^{\prime \prime \prime}+\left(y^{\prime \prime}\right)^{2} y^{\prime}+y=5\(. This is classified as a third-order nonlinear ODE.

Step-by-step solution

Identify and classify

The given equation is \(y^{\prime \prime} + 3y^{\prime} + 4y = 0\). Since the equation depends only on the function \(y\) and its first and second derivatives, it is an ordinary differential equation. It is a linear equation with the highest derivative being the second derivative, so it is a second-order linear ODE. (b)

Identify and classify

The given equation is \(y^{\prime \prime \prime} + 2y^{\prime \prime} + 5yy^{\prime} = x\). It is a third-order differential equation as the highest derivative is the third derivative. However, since the function \(y\) is multiplied by its derivative \(y^{\prime}\), the equation is a nonlinear ODE. (c)

Identify and classify

The given equation is \(y^{iv}+y=\sin x\). This differential equation involves the fourth derivative of the function \(y\). There are no nonlinear multiplications of \(y\), its derivatives, and the given function is linear. It is a fourth-order linear ODE. (d)

Identify and classify

The given equation is \(y^{\prime \prime}+(\cos x) y^{\prime}+x^{2} y=0\). This differential equation contains the second derivative and the function \(y(x)\). No nonlinear terms are involved. Thus, it is a second-order linear inhomogeneous ODE, due to the presence of \(x^2\) in the coefficient of \(y\). (e)

Identify and classify

The given equation is \(\left(y^{\prime \prime}\right)^{2}+y^{\prime}=e^{x}\). The highest derivative in this equation is the second derivative, and it is squared. This means that it is a nonlinear second-order ODE. (f)

Identify and classify

The given equation is \(yy^{\prime \prime} + 2xy^{\prime} + y = 0\). This is a second-order ODE as the highest derivative is the second derivative. However, the function \(y\) is multiplied with its own derivatives, so it is nonlinear. Therefore, this equation can be classified as a second-order nonlinear ODE. (g)

Identify and classify

The given equation is $y^{\prime \prime \prime}+\left(y^{\prime \prime}\right)^{2} y^{\prime}+y=5\(. This equation has the term \)\left(y^{\prime \prime}\right)^{2} y^{\prime}$ which signifies a nonlinear multiplication of derivatives. As the highest derivative involved is the third derivative, it can be classified as a third-order nonlinear ODE.

Key concepts

Linear Differential Equations

When we talk about linear differential equations, we're referring to equations where the dependent variable and its derivatives appear linearly. This means they are not raised to any power greater than one or multiplied together. An example of this is the equation, \(y'' + 3y' + 4y = 0\). Here, because every term involving the function \(y\) or its derivative is linear, this makes it a linear differential equation.

Linear differential equations have the form:

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

In this formula, each \(a_i(x)\) is a function of \(x\), and \(g(x)\) is known as the non-homogeneous term. If \(g(x)\) is zero, it is a homogeneous equation. If not, it is non-homogeneous.
Linear equations are particularly useful because they come with a rich set of solution techniques. Solutions are often easier to derive and can be expressed using well-known functions such as exponentials, sines, and cosines.

Nonlinear Differential Equations

Nonlinear differential equations are characterized by the occurrence of nonlinear terms of the dependent variable \(y\) and its derivatives. These can include products of derivatives, powers of derivatives greater than one, or other nonlinear phenomena.

An example from the problem is \(y''' + 2y'' + 5yy' = x\). Because of the product \(yy'\) which makes it nonlinear, solving these equations is often more complex and does not always yield to neat analytical solutions.

In general, a nonlinear differential equation does not adhere to the superposition principle. This means that a linear combination of solutions does not necessarily yield another solution. This added complexity often requires numerical methods or perturbative techniques to approximate solutions.

Second-Order Differential Equations

Second-order differential equations involve the second derivative of the dependent variable. They are crucial for modeling various physical systems, such as harmonic oscillators, circuits, and mechanical vibrations.

Take the equation \(y'' + \cos(x)y' + x^2y = 0\). This is a classic example of a second-order linear inhomogeneous ordinary differential equation (ODE).

• "Second-order" indicates that the highest derivative is the second derivative: \(y''\).
• "Linear" signifies that the equation can be written as a linear combination of \(y\), \(y'\), and \(y''\).

Such equations can generally be solved using characteristic equations, variation of parameters, or series solutions, depending on the problem structure.

Third-Order Differential Equations

A third-order differential equation involves the third derivative of the dependent variable, marking a higher complexity in analysis and solution. These are less frequently encountered than second-order ones but still important in certain scenarios such as in control theory and nonlinear dynamics.

From our problem set, the equation \(y''' + \left(y''\right)^2 y' + y = 5\) is a third-order nonlinear ODE.

• "Third-order" indicates the highest derivative is \(y'''\) which directly affects the equation's complexity.
• "Nonlinear" because terms like \(\left(y''\right)^2 y'\) involve products of derivatives, deviating from linear simplicity.

Solving third-order equations often requires sophisticated approaches. These might include numerical methods or special functions, especially in nonlinear cases where linear techniques are inadequate.