Search

11.4: First order ODEs
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/11%3A_Introduction/11.04%3A_First_order_ODEs
17.5: Chaos
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/17%3A_Nonlinear_Systems/17.05%3A_Chaos
Mathematical chaos is not really chaos, there is precise order behind the scenes. Everything is still deterministic. However a chaotic system is extremely sensitive to initial conditions. This also me...Mathematical chaos is not really chaos, there is precise order behind the scenes. Everything is still deterministic. However a chaotic system is extremely sensitive to initial conditions. This also means even small errors induced via numerical approximation create large errors very quickly, so it is almost impossible to numerically approximate for long times. This is large part of the trouble as chaotic systems cannot be in general solved analytically.
11.E: Introduction (Exercises)
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/11%3A_Introduction/11.E%3A_Introduction_(Exercises)
If $\vec{u} = (u_1,u_2,u_3)$ is a vector, we have the divergence $\nabla \cdot \vec{u} = \frac{\partial u_1}{\partial x} + \frac{\partial u_2}{\partial y} + \frac{\partial u_3}{\partial z}$ and cu...If $\vec{u} = (u_1,u_2,u_3)$ is a vector, we have the divergence $\nabla \cdot \vec{u} = \frac{\partial u_1}{\partial x} + \frac{\partial u_2}{\partial y} + \frac{\partial u_3}{\partial z}$ and curl $\nabla \times \vec{u} = \Bigl( \frac{\partial u_3}{\partial y} - \frac{\partial u_2}{\partial z} , ~ \frac{\partial u_1}{\partial z} - \frac{\partial u_3}{\partial x} , ~ \frac{\partial u_2}{\partial x} - \frac{\partial u_1}{\partial y} \Bigr)$ .
15.1: Boundary value problems
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/15%3A_Fourier_series_and_PDEs/15.01%3A_Boundary_value_problems
Before we tackle the Fourier series, we need to study the so-called boundary value problems (or endpoint problems).
15.E: Fourier Series and PDEs (Exercises)
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/15%3A_Fourier_series_and_PDEs/15.E%3A_Fourier_Series_and_PDEs_(Exercises)
These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engin...These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.
12.E: First order ODEs (Exercises)
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/12%3A_First_order_ODEs/12.E%3A_First_order_ODEs_(Exercises)
These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engin...These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.
7.2: Application of Eigenfunction Series
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/07%3A_Eigenvalue_problems/7.02%3A_Application_of_Eigenfunction_Series
The eigenfunction series can arise even from higher order equations.
11: Appendix A- Linear Algebra
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/11%3A_Appendix_A-_Linear_Algebra
10.3: Applications of Nonlinear Systems
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/10%3A_Nonlinear_Systems/10.03%3A_Applications_of_Nonlinear_Systems
In this section we will study two very standard examples of nonlinear systems. First, we will look at the nonlinear pendulum equation. We saw the pendulum equation's linearization before, but we n...In this section we will study two very standard examples of nonlinear systems. First, we will look at the nonlinear pendulum equation. We saw the pendulum equation's linearization before, but we noted it was only valid for small angles and short times. Now we will find out what happens for large angles. Next, we will look at the predator-prey equation, which finds various applications in modeling problems in biology, chemistry, economics and elsewhere.
12.7: Autonomous equations
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/12%3A_First_order_ODEs/12.07%3A_Autonomous_equations
We could draw the slope field, but it is easier to just look at the or , which is a simple way to visualize the behavior of autonomous equations. Armed with the phase diagram, it is easy to sketch the...We could draw the slope field, but it is easier to just look at the or , which is a simple way to visualize the behavior of autonomous equations. Armed with the phase diagram, it is easy to sketch the solutions approximately: As time $t$ moves from left to right, the graph of a solution goes up if the arrow is up, and it goes down if the arrow is down.
12: Appendix B- Table of Laplace Transforms
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/12%3A_Appendix_B-_Table_of_Laplace_Transforms
$\Gamma(t)=\int_{0}^{\infty}e^{-\tau}\tau^{t-1}d\tau ,\qquad\text{erf}(t)=\frac{2}{\sqrt{\pi}}\int_{0}^{t}e^{-\tau^{2}}d\tau , \qquad\text{erfc}(t)=1-\text{erf}(t).$ \(\frac{\omega}{s^{2}+\omega ^{2... $\Gamma(t)=\int_{0}^{\infty}e^{-\tau}\tau^{t-1}d\tau ,\qquad\text{erf}(t)=\frac{2}{\sqrt{\pi}}\int_{0}^{t}e^{-\tau^{2}}d\tau , \qquad\text{erfc}(t)=1-\text{erf}(t).$ $\frac{\omega}{s^{2}+\omega ^{2}}$ $\frac{1}{\sqrt{\pi t}}\text{exp}\left(\frac{-a^{2}}{4t}\right)\quad (a\geq 0)$ $\frac{1}{\sqrt{\pi t}}-ae^{a^{2}t}\text{erfc}(a\sqrt{t})\quad (a>0)$ $s^{3}G(s)-s^{2}g(0)-sg'(0)-g''(0)$ $s^{n}G(s)-s^{n-1}g(0)-\cdots -g^{(n-1)}(0)$ $(f\ast g)(t)=\int_{0}^{t} f(\tau )g(t-\tau )d\tau$

Search

Text Color

Text Size

Margin Size

Font Type

Support Center

How can we help?