Problem 124 - Calculus Volume 3

Question

Classify the differential equation. Determine the order, whether it is linear and, if linear, whether the differential equation is homogeneous or non homogeneous. If the equation is second-order homogeneous and linear, find the characteristic equation. $$ \frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+\sin ^{2}(t) y=e^{t} $$

Step-by-Step Solution

Answer

2nd order, linear, non-homogeneous differential equation.

Step 1: Identify the Order of the Differential Equation

The order of a differential equation is determined by the highest derivative present in the equation. In the equation \( \frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+\sin^{2}(t) y=e^{t} \), the highest derivative is \( \frac{d^{2} y}{d t^{2}} \), which is a second derivative. So, the order of the differential equation is 2.

Step 2: Determine if the Equation is Linear

A differential equation is linear if the dependent variable and its derivatives appear to the power of one (not multiplied by each other or any other functions of the independent variable) and additive to each other. The given equation is \( \frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+\sin^{2}(t) y=e^{t} \). This equation is linear because each term involving \( y \) and its derivatives is either multiplied by a function of \( t \) or a constant: they are not multiplied together or raised to any power other than one.

Step 3: Check if the Equation is Homogeneous

A linear differential equation is homogeneous if all terms are composed of the dependent variable \( y \) and its derivatives are equal to zero. In the equation \( \frac{d^{2} y}{d t^{2}}+t \frac{d y}{d t}+\sin^{2}(t) y=e^{t} \), the presence of \( e^{t} \) on the right hand side makes it non-homogeneous as it's not equal to zero.

Step 4: Characterize the Equation

Since the differential equation is second-order, linear, and non-homogeneous, we do not need to find a characteristic equation as this calculation is only relevant to homogeneous linear differential equations.

Key Concepts

Order of Differential Equation

When dealing with differential equations, the concept of "order" is crucial. The order of a differential equation is simply the highest order of derivative of the dependent variable present in the equation.
For example, in the equation \( \frac{d^{2} y}{d t^{2}} + t \frac{d y}{d t} + \sin^{2}(t) y = e^{t} \), the highest derivative is \( \frac{d^{2} y}{d t^{2}} \).
This means that the equation is of second order.
Understanding the order is important because it provides insight into the behavior and solution methods for the differential equation. Higher order equations often require more complex techniques to solve.
In general, always look for the highest derivative term to determine the order. This will guide you on how to approach solving the differential equation.

Linear Differential Equations

A differential equation can be classified as linear if it satisfies specific criteria. The hallmark of a linear differential equation is that it involves only the linear terms of the dependent variable and its derivatives.
In other words, each term must involve either a simple multiplication by a constant or a function of the independent variable, but not multiplication of derivatives together or raising them to powers other than one.
Consider the example: \( \frac{d^{2} y}{d t^{2}} + t \frac{d y}{d t} + \sin^{2}(t) y = e^{t} \). Each term with \( y \) and its derivatives is to the power of one, and none of them are multiplied together.
Thus, it qualifies as a linear differential equation.
Recognizing linear differential equations is essential because they often have well-established solution methods, making them easier to tackle than their nonlinear counterparts.

Homogeneous vs Non-Homogeneous Differential Equations

The distinction between homogeneous and non-homogeneous differential equations is another important concept in understanding these equations.
A linear differential equation is termed homogeneous if all of the terms involving the dependent variable and its derivatives sum to zero. Imagine if the equation looks like \( \text{terms} = 0 \).
In contrast, a non-homogeneous equation has additional terms that do not include the dependent variable. Such is the case in \( \frac{d^{2} y}{d t^{2}} + t \frac{d y}{d t} + \sin^{2}(t) y = e^{t} \).
The term \( e^{t} \) is not multiplied by \( y \) or its derivatives, making the equation non-homogeneous.
Knowing whether an equation is homogeneous or not will affect the methods used to find its solution. Non-homogeneous equations usually require extra steps, using complementary functions and particular solutions, to solve.