1.10. Summary

Try these first

1. 1.

   Define a first-order linear differential equation.

   Hint

   Think about the highest derivative involved.

2. 2.

   What is an integrating factor?

   Hint

   Consider how it changes the equation.

3. 3.

   What characterizes a linear differential equation?

   • It includes higher powers of y.
   • It includes y and its derivatives to the first power.
   • It has no derivatives.

   Hint

   Remember the definition of linearity.

4. 4.

   True or False: An integrating factor is always necessary for solving all first-order linear differential equations.

   • True
   • False

   Hint

   Consider situations where simpler methods apply.

5. 5.

   Given the non-homogeneous differential equation d²y/dx² - 4y = sin(x), derive the general solution.

   Hint

   Start by solving the complementary equation before tackling the particular solution.

6. 6.

   Solve the first-order linear DE dy/dx + 3y = x^2. What steps do you take?

   Hint

   Don't forget to integrate considering the initial condition if provided.