1. Linear Differential Equations

Try these first

1. 1.

   Define a differential equation.

   Hint

   Think about functions and their rates of change.

2. 2.

   What is the order of the equation d²y/dx² + y = 0?

   Hint

   Identify the highest derivative in the equation.

3. 3.

   What defines a linear differential equation?

   • Dependent variable is squared
   • Derivatives are to the first power
   • Variables are multiplied together

   Hint

   Focus on the definition of 'linearity'.

4. 4.

   True or False: A second-order linear DE can be homogeneous.

   • True
   • False

   Hint

   Remember the differences between homogeneous and non-homogeneous forms.

5. 5.

   Given the second-order DE d²y/dx² + 3dy/dx + 2y = 0, find the general solution.

   Hint

   Identify the roots using factoring.

6. 6.

   For the non-homogeneous equation d²y/dx² - y = e^x, derive the particular solution.

   Hint

   Use the method of undetermined coefficients.