Differential Equations (1) - NU

1. Introduction and ClassificationsDefinitions: Concepts of order, degree, linearity, and nonlinearity of differential equations.Solutions: Distinguishing between general solutions, particular solutions, and singular solutions.Initial Value Problems (IVPs): Understanding existence and uniqueness theorems.2. First-Order Ordinary Differential EquationsSeparable Equations: Separating variables to solve via direct integration.Homogeneous Equations: Utilizing substitutions (like y = vx) to transform equations into separable forms.Exact Equations: Checking exactness using partial derivatives and finding potential functions; solving non-exact equations via integrating factors.Linear Equations: Utilizing standard integrating factors \(I(x) = e^{\int P(x)dx}\) to solve first-order linear structures.Bernoulli Equations: Transforming nonlinear structures into linear forms using appropriate substitutions.3. Higher-Order Linear Differential EquationsHomogeneous Higher-Order Equations: Solving equations with constant coefficients using auxiliary (characteristic) equations (real, repeated, and complex roots).Non-Homogeneous Higher-Order Equations: Finding particular solutions using the Method of Undetermined Coefficients and the Variation of Parameters.Linear Independence: Evaluating fundamental solution sets using the Wronskian determinant.4. The Laplace TransformFoundations: Definition, existence conditions, and transforms of elementary functions.Properties: Shifting theorems, derivatives of transforms, and transforms of derivatives.Applications: Solving linear initial value problems directly using inverse Laplace transforms.