Question

Write the following first-order differential equations in standard form. $$ \frac{d y}{d t}=4 y+t y+\tan t $$

Short answer

The standard form is \( \frac{d y}{d t} - (4 + t)y = \tan t \).

Step-by-step solution

Identify the Original Equation

The given first-order differential equation is \( \frac{d y}{d t} = 4y + ty + \tan t \). Our goal is to rewrite it in a standard form used for first-order linear differential equations.

Combine Like Terms

Notice that the terms \(4y\) and \(ty\) both involve the dependent variable \(y\). Combine these terms to get \( (4 + t)y \). The equation is now \( \frac{d y}{d t} = (4 + t)y + \tan t \).

Rewrite in Standard Form

The standard form of a first-order linear differential equation is \( \frac{d y}{d t} + P(t) y = Q(t) \). Rearrange the equation to match this form: \( \frac{d y}{d t} - (4 + t)y = \tan t \).

Identify P(t) and Q(t)

From the standard form \( \frac{d y}{d t} - (4 + t)y = \tan t \), observe that \(P(t) = -(4 + t)\) and \(Q(t) = \tan t\).

Key concepts

Standard Form

First-order differential equations can often be complicated, but rewriting them in a standard form simplifies the analysis. This standard form, especially for linear differential equations, helps in identifying the components and understanding the equation structure.
For first-order linear differential equations, the standard form is given by: \ \[ \frac{d y}{d t} + P(t) y = Q(t) \]
This transformation allows us to define clearly how the dependent variable \( y \) changes with respect to the independent variable \( t \). Once in this form, it's easier to apply techniques such as the integrating factor method for solving the equation.
By rearranging the original equation \( \frac{d y}{d t} = 4y + ty + \tan t \) into its standard form \( \frac{d y}{d t} - (4 + t)y = \tan t \), we reveal two distinct functions: \( P(t) = -(4 + t) \) and \( Q(t) = \tan t \). These functions play critical roles in understanding the solution path and behavior of the differential equation.

Linear Differential Equations

Linear differential equations involve derivatives and terms that are exactly proportional to the dependent variable or its derivatives. In the realm of first-order equations, they take the form: \ \[ \frac{d y}{d t} + P(t) y = Q(t) \]
Here, the equation is said to be 'linear' because \( y \) and its derivatives appear to the first power and are not multiplied together. This form is one of the most important and widely studied kinds of differential equations due to its relative simplicity and ease of solution.
When an equation is linear, specific methods, for instance, the integrating factor method, can be employed to find solutions effectively. These methods hinge on the structure laid out by \( P(t) \) and \( Q(t) \), guiding the approach to forming the solution.
Linear equations are fundamental in modeling various real-world processes, where the rate of change of a quantity depends linearly on the quantity itself, such as in thermal cooling models, population growth models, and more.

Dependent Variable

In differential equations, the dependent variable is the function we're solving for. It depends on the independent variable through the relationship established by the differential equation itself. In our context, \( y \) is the dependent variable, as expressed in the equation \( \frac{d y}{d t} = 4y + ty + \tan t \).
The dependent variable defines how changes in the independent variable \( t \) affect \( y \). This relationship is transformed into derivatives and algebraic terms which the differential equation ties together.
Understanding the behavior of the dependent variable is crucial as it helps predict how a change in one quantity will affect another. For first-order differential equations like this one, the solution not only provides the expression for \( y \) but also highlights the interaction between \( P(t) \), \( Q(t) \), and \( y \).
The aim is often to determine \( y(t) \), the function of the independent variable, which provides insight into how the system or situation described by the equation behaves over time or with varying conditions.