Separable Equations and Example of Bernoulli Equation

2.3. Separable Equations

Introduction to Separable Equations

• Definition: A differential equation is defined as separable if it can be expressed in the form:

  f(y)dy = g(x)dx

Example 2.4.1: Solving the Bernoulli Equation

• The equation we will solve is given by:

  y' - y = xy^{2} ag{2.4.3}

Step 1: Identify the Equation Type

• The equation presented is a first-order Bernoulli differential equation, which has the general form:

  y' + P(x)y = Q(x)y^n

• In our example, we identify:

  • P(x) = -1
  • Q(x) = x
  • n = 2

• The term on the right-hand side includes $y^2$, thus confirming it's a Bernoulli's equation where $n
  eq 0, 1$.

Step 2: Making the Equation Separable

• To make the Bernoulli equation separable, we can divide through by $y^2$ (the nonlinear term) to reorganize the equation:

  rac{y'}{y^2} - rac{y}{y^2} = rac{x}{y^2}

• This simplifies to:

  y' rac{1}{y^2} - rac{1}{y} = x

Step 3: Substitute for the New Variable

• Introduce a substitution to solve the equation more easily. Let:

  v = rac{1}{y}

• Thus, we also have:

  y = rac{1}{v} ext{ and } y' = -rac{v'}{v^2}

Step 4: Rewrite the Equation

• Using the substitution in the equation, we transform the original equation into:

  -v' + rac{1}{v} = x

• Rearranging yields:

  v' = rac{1}{v} - x

Step 5: Implement Separation of Variables

• Now that we separated the variables, we can integrate:

  rac{dv}{rac{1}{v} - x} = dx

Step 6: Solve the Integral

• Integration of both sides leads to:

  ext{Left Side: } ext{integral of } rac{dv}{rac{1}{v} - x}

  • This will involve appropriate techniques of integration (substitution, partial fractions, etc.).

• The solution will also lead us toward a specific form to manage the integral on the left side.

Step 7: Back Substitute to Find y

• Once we have evaluated the integral, we can back-substitute $v = rac{1}{y}$ to find $y$ in terms of $x$.

Conclusion

• The example outlines the necessary steps to transform a Bernoulli differential equation into separable form, allowing for integration and solution. This technique is crucial in solving non-linear ordinary differential equations systematically.