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- https://math.libretexts.org/Courses/Coastline_College/Math_C185%3A_Calculus_II_(Tran)/05%3A_Introduction_to_Differential_Equations/5.02%3A_Basics_of_Differential_Equationsalculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x...alculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.
- https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/08%3A_Introduction_to_Differential_Equations/8.01%3A_Basics_of_Differential_EquationsCalculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(...Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2420_Calculus_II/04%3A_Introduction_to_Differential_Equations/4.01%3A_Basics_of_Differential_EquationsCalculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(...Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.
- https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_II_(Reed)/06%3A_Introduction_to_Differential_Equations/6.01%3A_Basics_of_Differential_EquationsCalculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(...Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.
- https://math.libretexts.org/Bookshelves/Calculus/Informal_Calculus_with_Applications_to_Biological_and_Environmental_Sciences_(Seacrest)/05%3A_Differential_Equations/5.03%3A_Introduction_to_Differential_EquationsOne general principle that often holds true for population is that the rate it is growing is proportional to the size of the population. Here, we see \({\color{red} f(x) = e^x + x + 1}\) is given, and...One general principle that often holds true for population is that the rate it is growing is proportional to the size of the population. Here, we see \({\color{red} f(x) = e^x + x + 1}\) is given, and we can compute this by taking the derivative of each piece. \[\begin{align*} {\color{blue} f'(x)} & = {\color{red} f(x)} - x \\ {\color{blue} e^x + 1} & = {\color{red} (e^x + x + 1)} - x \\ e^x + 1 & = e^x + 1 \end{align*}\]
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q2/04%3A_Introduction_to_Differential_Equations/4.02%3A_Basics_of_Differential_EquationsCalculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(...Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.
- https://math.libretexts.org/Courses/Mission_College/Mission_College_MAT_003B/04%3A_Introduction_to_Differential_Equations/4.01%3A_Basics_of_Differential_EquationsCalculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(...Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus_II__Integral_Calculus_._Lockman_Spring_2024/07%3A_Introduction_to_Differential_Equations/7.01%3A_Basics_of_Differential_EquationsCalculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(...Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.
- https://math.libretexts.org/Courses/Mission_College/MAT_3B_Calculus_II_(Kravets)/06%3A_Introduction_to_Differential_Equations/6.01%3A_Basics_of_Differential_EquationsCalculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(...Calculus is the mathematics of change, and rates of change are expressed by derivatives. Thus, one of the most common ways to use calculus is to set up an equation containing an unknown function y=f(x) and its derivative, known as a differential equation. Solving such equations often provides information about how quantities change and frequently provides insight into how and why the changes occur.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/04%3A_Power_series_methods/4.03%3A_Singular_Points_and_the_Method_of_FrobeniusWhile behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be luc...While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be lucky and obtain a power series solution using the method of the previous section, but in general we may have to try other things.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/02%3A_Applications_of_First_Order_Equations/2.04%3A_Mixing_Problems/2.4E%3A_Exercises_for_Section_2.4This page presents a collection of mixture problems about tanks and salt solutions, focusing on scenarios involving varying inflow and outflow rates. It explores how to calculate salt quantities in ta...This page presents a collection of mixture problems about tanks and salt solutions, focusing on scenarios involving varying inflow and outflow rates. It explores how to calculate salt quantities in tanks over time using differential equations and examines long-term concentration behaviors. The exercises encourage the use of numerical methods to verify predictions regarding salt concentration changes and derive specific functions for various conditions.
