Practice Solving The Homogeneous Part (6.2) - Non-Homogeneous Equations

Practice: Solving the Homogeneous Part

Lesson 6.2Non-Homogeneous Equations

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9 questions

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6 sample questions

Question 1.
What is the general form of the auxiliary equation for a second-order linear homogeneous differential equation?
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Remember, it relates to the coefficients of the differential equation.
Question 2.
What is the form of the complementary function if there are distinct real roots?
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Think about how you combine the solutions of each root.
3Question 3.
What form does the complementary function take for distinct real roots?
- y_h = (C_1 + C_2x)e^{r x}
- y_h = C_1 e^{r_1 x} + C_2 e^{r_2 x}
- y_h = e^{αx}(C_1 cos(βx) + C_2 sin(βx))
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Focus on how these solutions are articulated.
4Question 4.
True or False: The form of the complementary function when roots are complex involves trigonometric functions.
- True
- False
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Recall the connection of sin and cos with imaginary numbers.
Question 5.
Solve the differential equation d²y/dx² - 5dy/dx + 6y = 0 and find the complete complementary function.
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Start by setting the roots using the quadratic formula.
Question 6.
For the equation d²y/dx² + 4y = 0, find the roots and formulate the complementary function.
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Determine the imaginary components of the roots carefully.

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