Differential Equations: Complementary Functions

The video tutorial addresses solving a differential equation where the right-hand side is a multiple of a complementary function. Initially, a test function fails due to its form matching the homogeneous solution. The correct approach involves multiplying the test function by x and using the product rule to differentiate. This leads to the correct particular integral and general solution, emphasizing the importance of checking test functions against complementary functions.

What is the main issue with the right-hand side function in the differential equation?

• It is a multiple of a piece of the complementary function.

• It is a trigonometric function.

• It is a constant.

• It is a polynomial.

Why does the initial test function fail to solve the differential equation?

• It does not include a constant.

• It is too complex.

• It is identical to the complementary function.

• It is not a valid function.

What modification is suggested to the test function to make it work?

• Differentiate twice.

• Use a trigonometric function.

• Multiply by x.

• Add a constant.

Which rule is used for differentiating the modified test function?

• Power rule

• Chain rule

• Quotient rule

• Product rule

What is the value of the constant 'a' after solving the differential equation?

• -9

• 9

• 0

• 1

What is the particular integral of the differential equation?

• 9x e^2x

• -9 e^2x

• -9x e^2x

• 9 e^2x

Why is it important to find the complementary function first?

• To simplify calculations.

• To avoid using the same form in the test function.

• To ensure the equation is homogeneous.

• To eliminate constants.

What should be done if the test function is a piece of the complementary function?

• Use a different equation.

• Multiply the test function by x.

• Add a constant to the test function.

• Differentiate the test function.

What is the general solution to the differential equation?

• y = c1 e^2x + 9x e^2x

• y = c1 e^2x - 9x e^2x

• y = c1 e^2x + c2 e^2x - 9x e^2x

• y = c1 e^2x + c2 e^2x

What is the key takeaway from solving this differential equation?

• Always use the product rule.

• Find the complementary function and particular integral separately.

• Avoid using constants.

• Use a polynomial test function.