Practice Homogeneous Equations With Constant Coefficients (1.5) - Linear Differential Equations

Practice: Homogeneous Equations with Constant Coefficients

Lesson 1.5Linear Differential Equations

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8 questions

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Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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2 questions · about 1 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is the general form of a homogeneous equation?
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Remember it involves derivatives of the function y.
Question 2.
Define what an auxiliary equation is.
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Think of it as the characteristic polynomial derived from the differential equation.
3Question 3.
What form does a homogeneous differential equation take?
- a dy/dx + by = 0
- a d²y/dx² + b dy/dx + cy = 0
- b dy/dx + c = 0
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Consider the derivatives involved in the equation.
4Question 4.
True or false: The roots of the auxiliary equation can be complex.
- True
- False
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Think about the types of solutions we discussed.
Question 5.
Find the general solution for the differential equation d²y/dx² - 4dy/dx + 4y = 0 and discuss the significance of your results.
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Consider how repeated roots change the form of the solution.
Question 6.
Solve the equation d²y/dx² + 2dy/dx + 5y = 0, derive the general solution and reflect on its application context in engineering.
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Use the quadratic formula on the auxiliary equation to find roots.

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