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)dxf(y)dy = g(x)dx

Example 2.4.1: Solving the Bernoulli Equation

  • The equation we will solve is given by:

    y′−y=xy2ag2.4.3y' - 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)yny' + 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:

    y′y2−yy2=xy2\frac{y'}{y^2} - \frac{y}{y^2} = \frac{x}{y^2}

  • This simplifies to:

    y′1y2−1y=xy' \frac{1}{y^2} - \frac{1}{y} = x

Step 3: Substitute for the New Variable

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

    v=1yv = \frac{1}{y}

  • Thus, we also have:

    y=1vextandy′=−v′v2y = \frac{1}{v} ext{ and } y' = -\frac{v'}{v^2}

Step 4: Rewrite the Equation

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

    −v′+1v=x-v' + \frac{1}{v} = x

  • Rearranging yields:

    v′=1v−xv' = \frac{1}{v} - x

Step 5: Implement Separation of Variables

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

    dv1v−x=dx\frac{dv}{\frac{1}{v} - x} = dx

Step 6: Solve the Integral

  • Integration of both sides leads to:

    extLeftSide:extintegralofdv1v−xext{Left Side: } ext{integral of } \frac{dv}{\frac{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.