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- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.01%3A_The_Laplace_Transform/5.1E%3A_Exercises_for_Section_5.1This page outlines exercises on finding Laplace transforms and their inverses for various functions, including polynomials, trigonometric, and piecewise functions. It provides examples and answers for...This page outlines exercises on finding Laplace transforms and their inverses for various functions, including polynomials, trigonometric, and piecewise functions. It provides examples and answers for certain problems, illustrating the application of Laplace transform rules. The exercises emphasize techniques such as integration by parts and the use of properties related to exponential and trigonometric functions within the context of Laplace transforms.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.07%3A_Variation_of_Parameters/3.7E%3A_Exercises_for_Section_3.7This page outlines exercises focused on solving differential equations through the method of variation of parameters. It includes various equations, some with solutions, and tasks that involve applyin...This page outlines exercises focused on solving differential equations through the method of variation of parameters. It includes various equations, some with solutions, and tasks that involve applying this method to linear equations and initial value problems. Key mathematical principles, such as Abel's formula for Wronskian, are emphasized.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.03%3A_Trigonometric_SubstitutionThe technique of trigonometric substitution comes in very handy when evaluating integrals of certain forms. This technique uses substitution to rewrite these integrals as trigonometric integrals.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/04%3A_Parametric_Equations/4.02%3A_Calculus_of_Parametric_CurvesNow that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a gi...Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve?
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.09%3A_Calculus_of_the_Hyperbolic_FunctionsThis page discusses differentiation and integration of hyperbolic functions and their inverses, emphasizing their calculus applications, particularly in modeling catenary curves. Key objectives includ...This page discusses differentiation and integration of hyperbolic functions and their inverses, emphasizing their calculus applications, particularly in modeling catenary curves. Key objectives include understanding derivatives, integrals, and their respective formulas for hyperbolic functions, as well as domain considerations for inverse functions. The text provides examples, exercises, and instructions for evaluating integrals using u-substitution.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/03%3A_Techniques_of_Integration/3.05%3A_Other_Strategies_for_Integration/3.5E%3A_Exercises_for_Section_3.5This page presents exercises for evaluating integrals, featuring specific integral problems with solutions, including logarithmic, arctangent, and hyperbolic functions. It includes methods for calcula...This page presents exercises for evaluating integrals, featuring specific integral problems with solutions, including logarithmic, arctangent, and hyperbolic functions. It includes methods for calculating areas and volumes from curves, along with instructions for confirming results using calculators. The document is attributed to OpenStax and encompasses both definite and indefinite integrals.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.05%3A_Separable_EquationsThis page discusses separable differential equations, emphasizing identification, variable separation, and the integration process for solutions. It addresses implicit solutions, their importance, cha...This page discusses separable differential equations, emphasizing identification, variable separation, and the integration process for solutions. It addresses implicit solutions, their importance, challenges in deriving explicit forms, and includes examples demonstrating the method.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.02%3A_The_Method_of_Undetermined_Coefficients_IThis section present the method of undetermined coefficients, which can be used to solve nonhomogeneous equations of the form ay''+by'+cy=F(x) where a, b, and c are constants and F(x) has a special ...This section present the method of undetermined coefficients, which can be used to solve nonhomogeneous equations of the form ay''+by'+cy=F(x) where a, b, and c are constants and F(x) has a special form that is still sufficiently general to occur in many applications. This sections makes extensive use of the idea of variation of parameters introduced previously.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.07%3A_Existence_and_Uniqueness_of_Solutions_of_Nonlinear_EquationsAlthough there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximati...Although there are methods for solving some nonlinear equations, it is impossible to find useful formulas for the solutions of most. Whether we are looking for exact solutions or numerical approximations, it is useful to know conditions that imply the existence and uniqueness of solutions of initial value problems for nonlinear equations. In this section we state such a condition and illustrate it with examples.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/05%3A_Introduction_to_Differential_Equations/5.03%3A_Separable_EquationsWe now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of discipli...We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at the end of the section.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/00%3A_Front_Matter/05%3A_About_the_BookFrom understanding the rate of change to calculating areas and volumes of irregular shapes, this book encourages learners to explore the richness of Calculus in various fields, including physics, econ...From understanding the rate of change to calculating areas and volumes of irregular shapes, this book encourages learners to explore the richness of Calculus in various fields, including physics, economics, and engineering. The online version of the textbook includes 3D figures which gives a three-dimensional visual experience to the learner to understand the concept.
