Ordinary Differential Equations:Cheat Sheet/Second Order Homogeneous Ordinary Differential Equations

With Constant Coefficients General Form or , where is called the polynomial differential operator with constant coefficients. Solution Solve the auxiliary equation, , to get If are Real and distinct, then Real and equal, then Imaginary, , then Euler-Cauchy Equations General Form or where is called the polynomial differential operator. Solution Solving is equivalent to solving General Homogenous ODE with Variable Coefficients If one particular solution is known If one solution of a homogeneous linear second order equation is known, , original equation can be converted to a linear first order equation using substitutions and subsequent replacement . Abel's identity For the homogeneous linear ODE , Wronskian of its two solutions is given by Solution with Abel's identity Given a homogenous linear ODE and a solution of ODE, , find Wronskian using Abel’s identity and by definition of Wronskian, equate and solve for . Few Useful Notes If are linearly dependent, If , for some , then . First Order Ordinary Differential Equations Second Order Inhomogeneous Ordinary Differential Equations Ordinary Differential Equations:Cheat Sheet