Practice - Examples
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Practice Questions
Test your understanding with targeted questions
What is the convolution of f(t) = t and g(t) = e^{-t}?
💡 Hint: Use integration by parts to solve the integral.
How do we use the Laplace transform in solving differential equations?
💡 Hint: Identify the initial conditions before applying the transform.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the result of the convolution of f(t) = t and g(t) = e^{-t}?
💡 Hint: Think of the integral representation of convolution.
True or False: Convolution allows us to convert a differential equation into an algebraic equation.
💡 Hint: Consider the conversion process.
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Challenge Problems
Push your limits with advanced challenges
Evaluate the convolution of f(t) = sin(t) and g(t) = cos(t). Provide detailed steps showing the setup and final result.
💡 Hint: Use angles for simplifying the integration.
Solve the differential equation y'' + 2y' + y = e^{-t} using convolution and provide the solution expression.
💡 Hint: Express the right side with the impulse response for simplification.
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Reference links
Supplementary resources to enhance your learning experience.