1.4.1. General Form

Try these first

1. 1.

   What is the general form of a second-order linear differential equation?

   Hint

   Look for the highest derivative present in the equation.

2. 2.

   Define a homogeneous differential equation.

   Hint

   What happens to the non-homogeneous part?

3. 3.

   What is the standard form of a second-order linear differential equation?

   • $$\\frac{d^2y}{dx^2} + P(x) \\frac{dy}{dx} + Q(x)y = R(x)$$
   • $$\\frac{dy}{dx} + P(y) = R(x)$$
   • $$\\frac{d^2y}{dx^2} + Q(x) = 0$$

   Hint

   Look for the order of the derivatives present.

4. 4.

   Which type of equation is represented by $R(x) = 0$?

   • True
   • False

   Hint

   Remember what makes an equation homogeneous.

5. 5.

   For the differential equation $\frac{d^2y}{dx^2} + 5y = 0$, find the auxiliary equation and classify its solutions based on the roots.

   Hint

   What form does the solution take when roots are complex?

6. 6.

   Given the non-homogeneous equation $\frac{d^2y}{dx^2} - y = e^{2x}$, identify the general solution and the particular solution.

   Hint

   What’s your strategy for handling the right side of the equation?