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@@ -2,7 +2,21 @@
title: Differential Equations
---
# Differential equations
# Differential equations 1
The lecture on complex numbers consists of three parts, each with their own video:
- [First examples of differential equations](#first-examples-of-differential-equations-definitions-and-strategies)
- [Theory of systems of first-order differential equations](#theory-of-systems-of-differential-equations)
- [Solving homogeneous first-order differential equations with constant coefficients](#solving-homogeneous-linear-system-with-constant-coefficients)
**Total video length: 1 hour 15 minutes 4 seconds**
## First examples of differential equations: Definitions and strategies
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/IUr38H4dcWI?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
### Definitions
A differential equation or DE is any equation which involves both a function and some
derivative of that function. In this lecture we will be focusing on
@@ -80,7 +94,7 @@ $$\dot{\vec{x}(t)} = \vec{f}(\vec{x}(t),t) $$
with initial condition $\vec{x}(t_0) = \vec{x}_0$.
# Basic examples and strategies
### Basic examples and strategies
The simplest type of differential equation is the type learned about in the
integration portion of a calculus course. Such equations have the form,
@@ -237,7 +251,10 @@ we will develop the general theory for linear equations which will allow us to
tackle such higher order equations. For now, we move on to considering systems
of coupled first order linear DE's.
# Systems of first order differential equations
## Theory of systems of differential equations
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/4VoSMc08nQA?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
An intuitive presentation of a system of coupled first order differential
equations can be given by a phase portrait. Before demonstrating such a portrait,
@@ -269,9 +286,10 @@ a system of first order linear DE's
initial condition and the equation i.e. we know where each point "came from"
$\vec{x}(t'<t)$.
# Systems of linear first order differential equations
## Systems of linear first order differential equations
### Homogeneous systems
## Homogeneous systems ##
Any homogeneous system of first order linear DE's can be written in the form
$$\dot{\vec{x}} = A(t) \vec{x}, $$
@@ -309,11 +327,11 @@ $c_1, c_2, \cdots c_n$ are constants.
$\{\vec{\phi}_1 (t), \vec{\phi}_2(t), \cdots, \vec{\phi}_n (t) \}$ are a basis if and
only if they are linearly independent for fixed $t$:
$$det \big{(}\vec{\phi}_1 (t) | \vec{\phi}_2 (t) | \cdots | \vec{\phi}_n (t) \big{)} \neq 0.$$
$$\det \big{(}\vec{\phi}_1 (t) | \vec{\phi}_2 (t) | \cdots | \vec{\phi}_n (t) \big{)} \neq 0.$$
If this condition holds for one $t$, it holds for all $t$.
## Inhomogeneous systems ##
### Inhomogeneous systems
In addition to the homogeneous equation, an inhomogeneous equation has an
additional term, which may be a funcction of the independent variable.
@@ -400,7 +418,9 @@ $$\vec{\psi}(t)= \vec{\Phi}(t) \cdot \int \vec{\Phi}^{-1}(t) \vec{b}(t) dt .$$
$$\psi(t) = - \frac{a}{\lambda}. $$
## Homogeneous linear system with constant coefficients ##
## Solving homogeneous linear system with constant coefficients
<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/GGIDjgUpsH8?rel=0" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
The type of equaiton under consideration in this section looks like
@@ -467,7 +487,7 @@ solution are:
Next we will discuss how to determine a solution in practice, beyond the
formal solution just presented.
### Case 1: **A** diagonalizable ###
### Case 1: **A** diagonalizable
For an $n \times n$ matrix $A$, denote the $n$ distinct eigenvectors as
$\{\vec{v}_1, \cdots, \vec{v}_n \}$. By definition, the eigenvectors satisfy the
@@ -511,7 +531,7 @@ solutions $\vec{\phi}_{i}(t)$,
$$\vec{x}(t) = c_{1} e^{\lambda_1 t} \vec{v}_{1} + c_{2} e^{\lambda_2 t} \vec{v}_{2} + \cdots + c_{n} e^{\lambda_n t} \vec{v}_{n}.$$
and the vectors $\{e^{\lambda_1 t} \vec{v}_{1}, \cdots, e^{\lambda_n t} \vec{v}_{n} \}$
form a basis for the solution space since $det(\vec{v}_1 | \cdots | \vec{v}_n) \neq 0$
form a basis for the solution space since $\det(\vec{v}_1 | \cdots | \vec{v}_n) \neq 0$
(the $n$ eigenvectors are linearly independent).
!!! check "Example: Homogeneous first order linear system with diagonalizable constant coefficient matrix"
@@ -531,7 +551,7 @@ form a basis for the solution space since $det(\vec{v}_1 | \cdots | \vec{v}_n) \
To proceed following the solution technique, we determine the eigenvalues of
$A$,
$$det {\begin{bmatrix}
$$\det {\begin{bmatrix}
-\lambda & -1 \\
1 & - \lambda \\
\end{bmatrix}} = \lambda^2 + 1 = 0. $$
@@ -603,13 +623,13 @@ form a basis for the solution space since $det(\vec{v}_1 | \cdots | \vec{v}_n) \
\frac{1}{2}(e^{i t} + e^{-i t}) \\
\frac{1}{2 i}(e^{i t} - e^{-i t}) \\
\end{bmatrix} = \begin{bmatrix}
cos(t) \\
sin(t) \\
\cos(t) \\
\sin(t) \\
\end{bmatrix}. $$
### Case 2: **A** $2\times 2$, defective ###
### Case 2: **A** $2\times 2$, defective
In this case we consider the situation where $det(A- \lambda I)$
In this case we consider the situation where $\det(A- \lambda I)$
has a root $\lambda$ with multiplictiy 2, but only one eigenvector $\vec{v}_1$.
!!! check "Example: Matrix with eigenvalue of multiplicity 2 and only a single eigenvector."
@@ -623,7 +643,7 @@ has a root $\lambda$ with multiplictiy 2, but only one eigenvector $\vec{v}_1$.
The characteristic polynomial can be found by evaluating
$$det \big{(} \begin{bmatrix}
$$\det \big{(} \begin{bmatrix}
1-\lambda & 1 \\
0 & 1-\lambda \\
\end{bmatrix} \big{)} = 0$$
@@ -735,7 +755,8 @@ $$\vec{x}(t) = c_1 e^{\lambda_1 t} \vec{v}_1 + c_2(t e^{\lambda_1 t} \vec{v}_1
1 \\
\end{bmatrix}\big{)}.$$
### Bonus case 3: Higher multiplicity eigenvalues ###
### Bonus case 3: Higher multiplicity eigenvalues
In this case we consider the situation where the matrix $A$ has an
eigenvalue $\lambda$ with multiplicity $m>2$, and only one eigenvector $\vec{v}$
corresponding to $\lambda$, $(A - \lambda I)\vec{v}=0$. In this case notice
@@ -775,7 +796,7 @@ since we have demonstrated that $\vec{\phi}_{k}(t)$ is a solution of the DE.
(a) $\dot{x}(t) = t^4$
(b) $\dot{x}(t) = sin(t)$
(b) $\dot{x}(t) = \sin(t)$
2. [:grinning:] Solve, subject to the initial condition $x(0)=\frac{1}{2}$
@@ -787,15 +808,15 @@ since we have demonstrated that $\vec{\phi}_{k}(t)$ is a solution of the DE.
3. [:smirk:] Solve, subject to the given initial condition
(a) $\dot{x}(t)=-tan(x)sin(x)$, subject to $x(0)=1$.
(a) $\dot{x}(t)=-\tan(x)\sin(x)$, subject to $x(0)=1$.
(b) $\dot{x(t)}=\frac{1}{3} x^2+9$, subject to $x(0)=3$.
4. [:smirk:] Solve the following equation and list all possible solutions
$$\dot{x}=cos^2(x)$$
$$\dot{x}=\cos^2(x)$$
Hint: $\int \frac{1}{cos^2(x)} dx = tan(x) $
Hint: $\int \frac{1}{\cos^2(x)} dx = \tan(x) $
5. [:grinning:] Identify which of the following systems of equations is linear.
*Note thate you do not need to solve them!*
@@ -804,7 +825,7 @@ since we have demonstrated that $\vec{\phi}_{k}(t)$ is a solution of the DE.
$$\dot{x}_2 = x_1 x_2 - x_2$$
(b) $$\dot{x}_1 = e^{-t}x_1$$
$$\dot{x}_2 = \sqrt{t + cos(t)-1}x_1 + \frac{sin(t)}{t^2+t-1}x_2$$
$$\dot{x}_2 = \sqrt{t + \cos(t)-1}x_1 + \frac{\sin(t)}{t^2+t-1}x_2$$
(c) $$x^{(2)}_1 x_1 + \dot{x}_1 = 8 x_2$$
$$\dot{x}_2=5tx_2 + x_1$$