Lecture on Solving Differential Equations

Introduction to Differential Equations

• Differential Equation Example: ( \frac{dy}{dx} = \frac{4y}{x} )
• Solution Concept:
  • Solutions to differential equations are functions or sets of functions, not just values.

Testing for Solutions

• Method: Check if a function satisfies the differential equation by substituting and simplifying.

Example 1: ( y = 4x )

• Derivative Calculation:
  • ( \frac{dy}{dx} = 4 )
• Substitution:
  • Check if ( 4 = \frac{16x}{x} ) (i.e., ( 4 = 16 ))
• Conclusion: Not a solution because the equation is not satisfied.

Example 2: ( y = x^4 )

• Derivative Calculation:
  • ( \frac{dy}{dx} = 4x^3 )
• Substitution:
  • Check if ( 4x^3 = \frac{4x^4}{x} )
• Conclusion: Is a solution because the equation holds true.

Another Differential Equation

• Equation: ( f'(x) = f(x) - x )

Example 1: ( f(x) = 2x )

• Derivative Calculation:
  • ( f'(x) = 2 )
• Substitution:
  • Check if ( 2 = 2x - x )
• Conclusion: Not a solution because it does not hold for all ( x ).

Example 2: ( f(x) = x + 1 )

• Derivative Calculation:
  • ( f'(x) = 1 )
• Substitution:
  • Check if ( 1 = x + 1 - x )
• Conclusion: Is a solution because the equation holds for all ( x ).

Example 3: ( f(x) = e^x + x + 1 )

• Derivative Calculation:
  • ( f'(x) = e^x + 1 )
• Substitution:
  • Check if ( e^x + 1 = e^x + x + 1 - x )
• Conclusion: Is a solution because the equation is satisfied.

Summary

• Solutions to differential equations require functions that satisfy the equation for all values in their domain.
• Verification involves computing derivatives and substituting back into the original equation to check for equality.