Question

Explain your selections. $$ y^{\prime}=t \tan y $$

Short answer

The general solution is \( \ln|\sin(y)| = \frac{t^2}{2} + C \).

Step-by-step solution

Recognize the Type of Differential Equation

The equation \( y' = t \tan(y) \) is a first-order differential equation. More specifically, it can be classified as a separable differential equation because we can rearrange it to separate the variables \( t \) and \( y \).

Separate Variables

Rearrange the equation by dividing both sides by \( \tan(y) \) and multiplying both sides by \( dt \):\[ \frac{dy}{\tan(y)} = t \, dt \]. This separates \( y \) and \( t \) into different sides of the equation.

Integrate Both Sides

Integrate both sides of the equation. The left side requires integration of \( \frac{1}{\tan(y)} \), which is equivalent to \( \int \cot(y) \, dy \), and the right side is \( \int t \, dt \). Perform the integrations as follows:\[ \int \cot(y) \, dy = \ln|\sin(y)| + C_1 \]\[ \int t \, dt = \frac{t^2}{2} + C_2 \]

Solve for the General Solution

Combine the results of the integrals to form the general solution:\[ \ln|\sin(y)| = \frac{t^2}{2} + C \]where \( C = C_2 - C_1 \) is a constant of integration. This equation implicitly defines \( y \) in terms of \( t \), representing the general solution to the differential equation.

Key concepts

First-Order Differential Equations

Differential equations at their core are mathematical equations that involve derivatives, representing rates of change. First-order differential equations are the simplest type because they only involve the first derivative of the unknown function, usually denoted as \( y' \) or \( \frac{dy}{dt} \). These equations can take on many forms and behave in various ways depending on their structure. In the example provided, we deal with an equation \( y' = t \tan(y) \), which is first-order because it only includes the first derivative of \( y \).
The ultimate goal when working with these equations is to find a solution that satisfies the equation, typically expressed as a function of \( y \) in terms of the independent variable \( t \). This solution tells us how the dependent variable changes in response to the independent variable, described by the given relationship.

Integration Techniques

Integration is a fundamental mathematical technique that allows us to find functions from their derivatives. It's essential for solving differential equations as it helps us revert the differentiation process. When dealing with separable differential equations like \( y' = t \tan(y) \), we often rearrange the equation so that one side depends only on the dependent variable \( y \), and the other on the independent variable \( t \).

• Separating variables sometimes involves algebraic manipulation. For our example, we rewrite the equation as \( \frac{dy}{\tan(y)} = t \, dt \). This effectively isolates \( y \) and \( t \) each on different sides.
• Next, we integrate both sides separately. For the left side, \( \int \cot(y) \, dy \), which results in \( \ln|\sin(y)| + C_1 \), and for the right side, \( \int t \, dt \), which yields \( \frac{t^2}{2} + C_2 \).

These integrations help us build toward a solution, but the steps require knowledge of basic integral forms and rules. Successfully integrating both sides enables us to come closer to finding a relationship between \( y \) and \( t \) that satisfies the original differential equation.

General Solution

The general solution of a differential equation is a function or a set of functions that satisfy the given equation. After applying integration techniques, our goal is to find a general solution that connects the dependent and independent variables through an implicit or explicit expression.

• In the given example, after integrating, we arrive at the equation \( \ln|\sin(y)| = \frac{t^2}{2} + C \), where \( C \) is the constant of integration.
• This equation defines \( y \) implicitly in terms of \( t \), meaning \( y \) is not isolated on one side but is still defined within an equation that includes other terms involving \( t \).

The general solution thus embodies all possible solutions that solve the differential equation as variance in the constant \( C \) allows for different specific instances that fit initial conditions or additional constraints. A deeper understanding of the general solution can also help when applying it to real-world problems, where finding a specific solution matching given conditions is often necessary.