Solving Ordinary Differential Equations (ODEs): A Guide to Solving y' - y^2 1

In this article, we will guide you through the process of solving the first-order ordinary differential equation (ODE) y' - y^2 1. This equation is non-linear, meaning that it cannot be easily solved using the standard methods for linear ODEs. We will explore several techniques to find the general and particular solutions to this equation, including substitution methods and integration.

Understanding the Problem

Consider the ODE:

y' - y^2 1

where y' denotes the first derivative of y with respect to x.

Step-by-Step Solution

Step 1: Rewrite the Equation

First, let's rearrange the equation to a more manageable form:

y' y^2 1

Step 2: Use a Substitution

To proceed, we can use a substitution. Let:

y p

Then y' p' dp/dx. Substituting these into the equation, we get:

dp/dx p^2 1

Step 3: Express p in Terms of y

Next, we express in terms of y. Since y p, we have:

p y

Step 4: Separate Variables

Now we can separate the variables:

dp/(p^2 1) dx

Integrating both sides, we get:

∫ (1/(p^2 1)) dp ∫ dx

The left side integrates to arctan(p), and the right side integrates to x C, where C is the constant of integration. Therefore:

arctan(p) x C

Substituting back p y, we obtain:

arctan(y) x C

Step 5: Solve for y

To solve for y, we can take the tangent of both sides:

y tan(x C)

This is the general solution to the ODE. The constant C can be determined from initial conditions if provided.

Conclusion

The general solution to the ODE y' - y^2 1 is y tan(x C), where C is an arbitrary constant. This process demonstrates the step-by-step approach to solving non-linear ODEs, including substitution, variable separation, and integration.