Differential Equations - Separable Equations

Introduction

• Focus on nonlinear first order differential equations.
• First type: Separable Differential Equations.

Definition of Separable Differential Equations

• Can be expressed in the form: ( N(y) \frac{dy}{dx} = M(x) ).
• Condition: All ( y )'s must be with the derivative, and all ( x )'s must be on the other side.

Solving Separable Differential Equations

1. Integrate both sides with respect to ( x ):
   • ( \int N(y) \frac{dy}{dx} dx = \int M(x) dx ).
2. Use substitution:
   • Let ( u = y(x) ), hence ( du = y'(x) dx = \frac{dy}{dx} dx ).
   • Substitute into the integral: ( \int N(u) du = \int M(x) dx ).
3. Integration and Back-substitution:
   • Integrate both sides and substitute back for ( u ) to find ( y(x) ).
   • Note: Integration might not always be possible.

Practical Approach

• "Separate the derivative": Pretend you can write ( N(y) dy = M(x) dx ).
• Integrate: ( \int N(y) dy = \int M(x) dx ).
• Compare with the substitution method to confirm results.

Solution Types

• Implicit Solution: Not in form ( y = y(x) ).
• Explicit Solution: Solved for ( y(x) ).
• Validity: Check interval of validity for the solution.
  • Avoid division by zero, complex numbers, logarithms of negative/zero.

Examples

• Various solved examples illustrating the process and issues:
  • Example 1: Basic separable equation solved.
  • Example 2 to 6: Initial Value Problems (IVPs) solved, highlighting interval of validity and difficulties in obtaining explicit solutions.

Conclusion

• Be aware that not all separable differential equations yield explicit solutions.
• Check the range for which the solution is valid.

Additional Information

• Contact for assistance and additional resources on the topic.
• Content provided by Paul Dawkins, last modified on 2/6/2023.

This summary captures the key points and methodologies surrounding separable differential equations as explained by Paul Dawkins.