91Ó°ÊÓ

A separable differential equation is a type of first-order ordinary differential equation (ODE) that can be manipulated so that all terms involving the dependent variable (y) are on one side of the equation, and all terms involving the independent variable (t) are on the other side.
In the given exercise, the equation is \(\frac{dy}{dt} = \frac{y+1}{t+1}\).
We identify that this ODE is separable, because we can rewrite it to separate the variables as follows:

• Move y-related terms to one side: \(\frac{dy}{y+1}\).
• Move t-related terms to the other side: \(\frac{dt}{t+1}\).

By successfully doing this, we convert the single differential equation into two integrable functions.This technique is particularly useful as it simplifies the process of solving the ODE.
When you separate variables, it paves the way to use integration on both sides, ultimately finding a solution that ties y and t together.

Integration is a fundamental part of solving separable differential equations. After separating variables, the next step involves integrating both sides of the equation to find a relation between y and t.
Here’s how to proceed with the given exercise:

• First, we have: \(\frac{dy}{y+1} = \frac{dt}{t+1}\).
  
• To integrate, use the substitution method:
• Let \(u = y + 1\) on the left side, then \(du = dy\).
• Let \(v = t + 1\) on the right side, then \(dv = dt\).
• The integrals become: \( \int \frac{1}{u} \, du = \int \frac{1}{v} \, dv\).
• Applying the integration technique, we get:
  \( \ln|u| = \ln|v| + C \).
  

The integration for \( \frac{1}{u} \) results in the natural logarithm, \(\ln|u|\), and similarly for \(\frac{1}{v}\).
Understanding these common integrals is crucial for tackling differential equations efficiently.

An ordinary differential equation (ODE) is an equation that contains a function of one independent variable and its derivatives. The term 'ordinary' differentiates it from partial differential equations, which may involve multiple independent variables.
In the context of this problem, the given ODE is a first-order ODE because it involves the first derivative of y with respect to t, i.e., \( \frac{dy}{dt} = \frac{y+1}{t+1}\).

To solve this type of ODE, we typically follow these steps:

• Identify the type of ODE. In this case, it is separable.
• Separate the variables.
• Integrate both sides.
• Combine like terms and solve for the dependent variable (y).
• Determine the constant of integration (C), using any initial conditions provided.

The last step, determining the constant, is essential because the integration process results in a family of solutions, and an initial condition helps to pinpoint the exact solution.
Understanding the nature of ODEs and the standard methods to solve them forms the foundation of many applications in physics, engineering, and other scientific fields.