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https://math.libretexts.org/Bookshelves/Differential_Equations/A_Second_Course_in_Ordinary_Differential_Equations%3A_Dynamical_Systems_and_Boundary_Value_Problems_(Herman)/zz%3A_Back_Matter/10%3A_Index
Back Matter
https://math.libretexts.org/Bookshelves/Differential_Equations/A_Second_Course_in_Ordinary_Differential_Equations%3A_Dynamical_Systems_and_Boundary_Value_Problems_(Herman)/zz%3A_Back_Matter
8: Green's Functions
https://math.libretexts.org/Bookshelves/Differential_Equations/A_Second_Course_in_Ordinary_Differential_Equations%3A_Dynamical_Systems_and_Boundary_Value_Problems_(Herman)/08%3A_Green's_Functions
The function $G(x, \xi)$ is referred to as the kernel of the integral operator and is called the Green's function. The history of the Green's function dates back to 1828 , when George Green publishe...The function $G(x, \xi)$ is referred to as the kernel of the integral operator and is called the Green's function. The history of the Green's function dates back to 1828 , when George Green published work in which he sought solutions of Poisson's equation $\nabla^{2} u=f$ for the electric potential $u$ defined inside a bounded volume with specified boundary conditions on the surface of the volume.
3.1: Introduction
https://math.libretexts.org/Bookshelves/Differential_Equations/A_Second_Course_in_Ordinary_Differential_Equations%3A_Dynamical_Systems_and_Boundary_Value_Problems_(Herman)/03%3A_Nonlinear_Systems/3.01%3A_Introduction
Typically, we have a certain population, $y(t)$ , and the differential equation governing the growth behavior of this population is developed in a manner similar to that used previously for mixing pr...Typically, we have a certain population, $y(t)$ , and the differential equation governing the growth behavior of this population is developed in a manner similar to that used previously for mixing problems. We note that the rate of change of the population is given by the Rate In minus the Rate Out. Here we have denoted the birth rate as $b$ and the mortality rate as $m$ , . This gives the rate of change of population as
7.4: Bifurcations for First Order Equations
https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/07%3A_Nonlinear_Systems/7.04%3A_Bifurcations_for_First_Order_Equations
Figure $\PageIndex{2}$ : (a) The typical phase lines for $y' = y^2 − \mu$ . (b) Bifurcation diagram for $y' = y^2 − \mu$ . Figure $\PageIndex{4}$ : (a) Collection of phase lines for \(y^{\prime}=y...Figure $\PageIndex{2}$ : (a) The typical phase lines for $y' = y^2 − \mu$ . (b) Bifurcation diagram for $y' = y^2 − \mu$ . Figure $\PageIndex{4}$ : (a) Collection of phase lines for $y^{\prime}=y^{2}-\mu y$ . (b) Bifurcation diagram for $y^{\prime}=y^{2}-\mu y$ . The left one corresponds to $\mu<$ 0 and the right phase line is for $\mu>0$ . (b) Bifurcation diagram for $y^{\prime}=y^{3}-\mu y$ .
5.6: Systems of ODEs
https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/05%3A_Laplace_Transforms/5.06%3A_Systems_of_ODEs
LAPACE TRANSFORMS ARE ALSO USEFUL for solving systems of differential equations. We will study linear systems of differential equation in Chapter 6. For now, we will just look at simple examples of th...LAPACE TRANSFORMS ARE ALSO USEFUL for solving systems of differential equations. We will study linear systems of differential equation in Chapter 6. For now, we will just look at simple examples of the application of Laplace transforms.
5.1: The Laplace Transform
https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/05%3A_Laplace_Transforms/5.01%3A_The_Laplace_Transform
Up to this point we have only explored Fourier exponential transforms as one type of integral transform. The Fourier transform is useful on infinite domains. However, students are often introduced to ...Up to this point we have only explored Fourier exponential transforms as one type of integral transform. The Fourier transform is useful on infinite domains. However, students are often introduced to another integral transform, called the Laplace transform, in their introductory differential equations class. These transforms are defined over semi-infinite domains and are useful for solving initial value problems for ordinary differential equations.
8.4: Derivatives
https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/08%3A_Appendix_Calculus_Review/8.04%3A_Derivatives
Now that we know some elementary functions, we seek their derivatives. We will not spend time exploring the appropriate limits in any rigorous way. We are only interested in the results. We expect tha...Now that we know some elementary functions, we seek their derivatives. We will not spend time exploring the appropriate limits in any rigorous way. We are only interested in the results. We expect that you know the meaning of the derivative and all of the usual rules, such as the product and quotient rules.
3.5.2: Extreme Sky Diving
https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/03%3A_Numerical_Solutions/3.05%3A_Numerical_Applications/3.5.02%3A_Extreme_Sky_Diving
ON OCTOBER 14, 2012 FELIX BAUMGARTNER JUMPED from a helium balloon at an altitude of 39045 m(24.26mi or 128100ft) . According preliminary data from the Red Bull Stratos Mission 1 , as of November ...ON OCTOBER 14, 2012 FELIX BAUMGARTNER JUMPED from a helium balloon at an altitude of 39045 m(24.26mi or 128100ft) . According preliminary data from the Red Bull Stratos Mission 1 , as of November 6,2012 Baumgartner experienced free fall until he opened his parachute at 1585 m after 4 minutes and 20 seconds.
3.5.3: The Flight of Sports Balls
https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/03%3A_Numerical_Solutions/3.05%3A_Numerical_Applications/3.5.03%3A_The_Flight_of_Sports_Balls
ANOTHER INTERESTING PROBLEM IS THE PROJECTILE MOTION OF A SPORTS ball. In an introductory physics course, one typically ignores air resistance and the path of the ball is a nice parabolic curve. Howev...ANOTHER INTERESTING PROBLEM IS THE PROJECTILE MOTION OF A SPORTS ball. In an introductory physics course, one typically ignores air resistance and the path of the ball is a nice parabolic curve. However, adding air resistance complicates the problem significantly and cannot be solved analytically. Examples in sports are flying soccer balls, golf balls, ping pong balls, baseballs, and other spherical balls.
6.5: Solving Constant Coefficient Systems in 2D
https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/06%3A_Linear_Systems/6.05%3A_Solving_Constant_Coefficient_Systems_in_2D
Before proceeding to examples, we first indicate the types of solutions that could result from the solution of a homogeneous, constant coefficient system of first order differential equations.

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