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- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.03%3A_ConvolutionThis page discusses the use of inverse Laplace transforms and convolution in solving ordinary differential and Volterra integral equations. It highlights the simplification of computations through the...This page discusses the use of inverse Laplace transforms and convolution in solving ordinary differential and Volterra integral equations. It highlights the simplification of computations through these methods. An example is provided where a differential equation involving an integral is transformed into the frequency domain, resulting in the expression \( X(s) = \dfrac{s-1}{s^2-2} \). The final solution is obtained as \( x(t) = \cosh(\sqrt{2}\, t) - \dfrac{1}{\sqrt{2}} \sinh(\sqrt{2}\,t) \).
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEsIn this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different f...In this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. Here we concentrate primarily on second-order equations with constant coefficients.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.02%3A_Transforms_of_Derivatives_and_ODEs/5.2E%3A_Exercises_for_Section_5.2This page covers exercises on Laplace transforms and differential equations, emphasizing problem-solving techniques, verification of transforms, piecewise function formulation, and second-order differ...This page covers exercises on Laplace transforms and differential equations, emphasizing problem-solving techniques, verification of transforms, piecewise function formulation, and second-order differential equations with varying conditions. It explores properties like differentiation under the integral sign and the second shifting property, and requires finding transfer functions. Solutions for specific cases are also included in the exercises.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEsThis page introduces systems of ordinary differential equations (ODEs), highlighting how a single equation can lead to multiple dependent variables. It discusses linear systems of ODEs and the eigenva...This page introduces systems of ordinary differential equations (ODEs), highlighting how a single equation can lead to multiple dependent variables. It discusses linear systems of ODEs and the eigenvalue method for solving linear homogeneous constant coefficient systems, including exercises for practice. The content also covers the role of matrices within these systems, indicating that it is part of a broader educational guide funded by NSF grants.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEs/6.02%3A_Linear_systems_of_ODEsThis page summarizes concepts related to homogeneous and nonhomogeneous systems of ordinary differential equations (ODEs). It explains differentiation rules for matrix and vector-valued functions, sol...This page summarizes concepts related to homogeneous and nonhomogeneous systems of ordinary differential equations (ODEs). It explains differentiation rules for matrix and vector-valued functions, solving first-order linear systems, superposition principles, and linear independence.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEs/6.01%3A_Introduction_to_Systems_of_ODEs/6.1E%3A_Exercises_for_Section_6.1This page focuses on exercises involving systems of differential equations, emphasizing finding general solutions, transforming higher-order ODEs into first-order systems, and solving specific equatio...This page focuses on exercises involving systems of differential equations, emphasizing finding general solutions, transforming higher-order ODEs into first-order systems, and solving specific equations with initial conditions. It encourages the application of differential equation techniques and requires explicit answers that include constants, functions, and initial values, aiming to enhance skills in solving and manipulating these equations systematically.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.02%3A_Transforms_of_Derivatives_and_ODEsThe procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable t . We apply the Laplace transform to transform the equation into ...The procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable t . We apply the Laplace transform to transform the equation into an algebraic (non differential) equation in the frequency domain. We solve the equation for X(s) . Then taking the inverse transform, if possible, we find x(t). Unfortunately, not every function has a Laplace transform, not every equation can be solved in this manner.
