Question

Classify the following equations, specifying the order and type (linear or non-linear): (a) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{2}\) (b) \(\frac{\mathrm{d} y}{\mathrm{~d} t}+\cos y=0\)

Short answer

Question: Classify the given equations in terms of order and type (linear or non-linear): (a) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{2}\) (b) \(\frac{\mathrm{d} y}{\mathrm{~d} t}+\cos y=0\) Answer: (a) Linear second-order equation, (b) Non-linear first-order equation.

Step-by-step solution

(Classification of Equation (a))

To classify equation (a), \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{2}\), first identify the highest derivative, which in this case is the second derivative (\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\)). Therefore, the order of this equation is 2. Next, determine if the equation is linear or non-linear. This equation is linear since the dependent variable and its derivatives appear only to the first power and are not multiplied together. So, equation (a) is a linear second-order equation.

(Classification of Equation (b))

To classify equation (b), \(\frac{\mathrm{d} y}{\mathrm{~d} t}+\cos y=0\), first identify the highest derivative, which in this case is the first derivative (\(\frac{\mathrm{d} y}{\mathrm{~d} t}\)). Therefore, the order of this equation is 1. Next, determine if the equation is linear or non-linear. This equation is non-linear because of the presence of the term \(\cos y\), which makes the equation not fit the linear equation description. So, equation (b) is a non-linear first-order equation.

Key concepts

Linear Equations

Linear equations in differential equations are those where the unknown function and its derivatives appear linearly. This means that each term is either a constant or a product of a constant and the first power of the dependent variable or its derivative.
A simple way to verify if a differential equation is linear is to check:

• The dependent variable and its derivatives are not multiplied by each other.
• There are no functions of the dependent variable other than itself appearing in the equation.
• The derivatives don't appear in exponents, denominators, or within non-linear functions.

A standard form of a linear differential equation is \( a_n(x)\frac{d^n y}{dx^n} + a_{n-1}(x)\frac{d^{n-1} y}{dx^{n-1}} + ... + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x) \), where each \( a_i(x) \) is a coefficient that can be a function of \( x \), but not \( y \).In the provided exercise, equation (a) is linear as it satisfies these conditions. The absence of any terms involving non-linear functions of \( y \,\ \frac{dy}{dx} \,\text{or}\ \frac{d^2y}{dx^2} \) verifies its linearity.

Non-linear Equations

Non-linear differential equations differ from linear ones in that they involve non-linear terms of the unknown function or its derivatives. This could mean:

• A derivative is raised to a power other than one.
• Products of the function or its derivatives.
• Non-linear functions of the dependent variable, such as exponential, trigonometric, or logarithmic functions.

Non-linear equations are more complex because they can demonstrate behaviors such as bifurcations, limit cycles, or chaos. Unlike linear equations, these challenges make analytical solutions often unavailable or difficult to obtain.In the example of equation (b) from the exercise, the presence of \( \cos y \) makes it non-linear. This term introduces non-linearity because it is a trigonometric function of the dependent variable, \( y \), thus breaking the criteria for linearity.

Order of Equations

The order of a differential equation is determined by the highest derivative present in the equation. Recognizing the order is crucial because it influences both the methods used for solving the equation and the nature of the solutions.

• If a differential equation includes a second derivative, it's said to be a second-order equation.
• A first derivative as the highest term indicates a first-order equation.
• The order is not affected by the coefficients or the independent variables.

In the exercise solved here, equation (a) is a second-order equation due to the presence of the second derivative, \( \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}} \), while equation (b) is first-order because \( \frac{\mathrm{d} y}{\mathrm{~d} t} \) is the highest derivative term.Understanding the order helps to better predict the behavior of the solution, such as the number of initial conditions needed to form a complete solution, and potential complexities in solving the equation.