Question

\(y y^{\prime}+t y=0\).

Short answer

The solutions are \( y = 0 \) and \( y = -\frac{t^2}{2} + C \).

Step-by-step solution

Identify Given Variables and Simplify

The differential equation provided is: \[ y \frac{dy}{dt} + t y = 0. \]Our goal is to solve for \( y \). Start by recognizing that \( y y' \) represents \( y \frac{dy}{dt} \). Factor \( y \) from the terms to rewrite the equation as:\[ y \left( \frac{dy}{dt} + t \right) = 0. \]

Set Up the Equation for Separation

Assuming \( y eq 0 \), divide through by \( y \):\[ \frac{dy}{dt} + t = 0. \]This results in a simpler, separable differential equation.

Solve the Homogeneous Differential Equation

Integrate both sides with respect to \( t \):\[ \int \frac{dy}{dt} \, dt = \int -t \, dt. \]This simplifies to:\[ y = -\frac{t^2}{2} + C, \]where \( C \) is the constant of integration.

Consider Solution for y = 0

Since we initially assumed \( y eq 0 \), we should consider the trivial solution where \( y = 0 \). Substitute \( y = 0 \) into the original equation to confirm it satisfies the equation.

Key concepts

Separation of Variables

Separation of variables is a technique used to solve differential equations, where the goal is to separate the variables involved so that each side of the equation contains only one variable. In this method, we often move all terms involving one variable to one side of the equation and terms involving the other to the opposite side. This allows us to more easily integrate both sides.

For example, consider the differential equation given as \( y \frac{dy}{dt} + t y = 0 \). We can rewrite it by factoring out \( y \) to become \( y(\frac{dy}{dt} + t) = 0 \). Assuming \( y eq 0 \), we can divide through by \( y \), giving us \( \frac{dy}{dt} = -t \).

This adjustment transforms the equation into a simpler form, ideal for the separation of variables method. By isolating \( \frac{dy}{dt} \) on one side and \(-t\) on the other, we can integrate both sides:

• Integrating the left side with respect to \( t \), we have \( \int \frac{dy}{dt} \, dt \).
• Integrating the right side, gives us \( \int -t \, dt \).

Thus, we have set the stage for solving the equation after separation.

Homogeneous Differential Equation

A homogeneous differential equation is one in which all the terms are a function of a combination of the dependent and independent variables. In simple terms, it is an equation set to zero where each term contains the unknown function or its derivatives.

In our example, the equation \( y \frac{dy}{dt} + t y = 0 \) is a homogeneous differential equation. To solve it, we can rely on the fact that all terms involve the variable \( y \) and its derivative. By assuming \( y eq 0 \) and dividing through by \( y \), the equation becomes \( \frac{dy}{dt} = -t \), which allows us to proceed with solving.

This step transforms the equation into a form that can be integrated directly. Homogeneous differential equations often require some manipulation, such as factoring or dividing, to make them suitable for techniques like separation of variables.

Constant of Integration

The constant of integration is an important concept that arises when solving differential equations involving integration. When we integrate a function, there can be an infinite number of solutions, each differing by a constant amount. This constant is what we call the constant of integration, typically represented by \( C \).

In our worked example, after performing the integration:

• For the left side, \( \int \frac{dy}{dt} \, dt \) simplifies to \( y \).
• For the right side, \( \int -t \, dt \) simplifies to \(-\frac{t^2}{2} + C\).

Hence, the solution becomes \( y = -\frac{t^2}{2} + C \). The constant \( C \) represents all the possible shifts in the \( y \)-axis that the solution can take within the equation's constraints.

Whenever you solve a differential equation by integration, always remember to accommodate the constant of integration, as it affects the general solution.