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Understanding and Solving Differential Equations
Dec 7, 2024
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.
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