MATH2010 Cheat Sheet Contents MATH2010 Cheat Sheet. 1 Systems of ODE's 2 Classifying ODE's. 2 Linear The Superposition Principle. 2 Solving Systems of 2 coupled 1st order ODE's. 2 Phase Portraits with Eigenvalues 4 2 Critical Points and Stability. 5 Laplace Transforms. 7 Basic Transformations. 7 Inverse Laplace Transforms 7 The Gamma Function and L(ta) where a is not an integer 8 When is L(f(t)) well defined 8 Solving ODE's Using the Laplace Transform 8 The Second Shifting Theorem. 9 Laplace Transform of Dirac Delta Function: 10 Convolution Theorem 10
Systems of ODE's An ODE is a function with one independent variable. Classifying ODE's The order of an ODE is the order of its highest derivative. Linear An ODE is linear if it is linear in the unknown and its derivatives, but it doesn't need to be linear in the independent variable. For example: is linear is not linear is not linear A linear ODE is either homogeneous or inhomogeneous. An ODE is homogeneous if there are no terms without the dependant variable, . When is a solution, then so is for any constant . The Superposition Principle All linear, homogeneous ODE's obey. You can take linear combinations of two known solutions to form new solutions. So if and are solutions, then is also a solution for any constants and . This also works for systems. If and are solutions to a 2D linear homogeneous system, then for any constants and , is also a solution. The superposition principle ONLY applies to Linear homogeneous systems. Solving Systems of 2 coupled 1st order ODE's Any second order linear ODE can be written as a system of two coupled 1st order ODE's. Let: Therefore:
The homogeneous case with constant coefficients: This is called the EIGENVALUE equation for A. Solve for eigenvalues of A. If there are two eigenvalues, and with eigenvectors and then: and are solutions to the eigenvalue equation for A. The GENERAL SOLUTION to the matrix equation is a linear combination of these: (From the superposition principle) for any constants and . Provided that the two solutions and are linearly independent, the solution can be written in the matrix form: The matrix: is called the fundamental matrix. If the determinant of this matrix (often called the Wronskian) is non-zero, then and are linearly independent and the general solution to this matrix equation is given by: Where: and