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Homogeneous Linear Equations with Constant Coefficients A second order homogeneous equation with constant coefficients is written as where a, b and c are constant. This type of equation is very useful in many applied problems (physics, electrical engineering, etc..). Let us summarize the steps to follow in order to find the general solution: Write down the characteristic equation This is a quadratic equation. Let If and and be its roots we have ; are distinct real numbers (this happens if ), then the general solution is (3) If If (which happens if and ), then the general solution is are complex numbers (which happens if ), then the general solution is where , that is, Example: Find the solution to the IVP Solution: Let us follow the steps: Characteristic equation and its roots Since 4-8 = -4<0, we have complex roots General solution . Therefore, and ; ; In order to find the particular solution we use the initial conditions to determine and . First, we have . Since , we get From these two equations we get , which implies