For the special case $$\frac{dy}{dt}=f(y)$$ where $f(c)=0$ for c=constant, these values are called critical points or equilibrium points and represent limiting values of the integral curves as $t \rightarrow \infty$.
Make sure you can plot the direction field by hand, sketch solution curves, and describe the behavior of integral curves.
Make sure you include the integration constant...
| ODE | $dy/dt+p(t)y=g(t)$ | $dy/dt=f(t,y)$ | Existence | Existence and Uniqueness for any interval on which $p(t)$ and $g(t)$ are continuous. | Continuity of $f$ and $\partial f/\partial y$ imply local existence and uniqueness. |
| Type of Solution | Eplicit (using integration factors) | Implicit (or no analytical solution possible) |
| Numerical Solution | Euler's method, fixed step size | Implicit methods, variable step size |
| Graphical Solution, Direction Field | Solution bounded if $p(t)$ and $g(t)$ are. | Solutions can "blow up" at finite time |
Almost all differential equations modeling real physical problems are non linear. Linear equations occur when processes are simplified, or non linear terms are neglected. Singularities (i.e. infinite behavior) does not occur in nature because terms that are small away from the singularity begin to dominate near the singularity, which necessitates a more complicted modeling process.
One-species population models are of this form, including the logistic equation $$\frac{dy}{dx}=f(y)=r(1-\frac{y}{K})y$$ or logistic equation with threshold $$\frac{dy}{dx}=f(y)=-r(1-\frac{y}{T})(1-\frac{y}{K})y$$
Equilibria ($f(y*)=0$) can be steady or unsteady. depending on the sign of $f'(y*)$.
The system of equations $$ a_{11}x_1+a_{12}x_2=b_1 $$ $$ a_{21}x_1+a_{22}x_2=b_2 $$ can be written as the matrix equation `[[a_{11},a_{12}],[a_{21},a_{22}]]` `[[x_1],[x_2]]`=`[[b_1],[b_2]]`
or Ax=b. If $\det(A)\ne 0$, then the inverse of $A$ exists, and there is a unique solution
Solutions for a 2x2 system can be found by either elimination, or row reduction. Cramer's rule can also be used.
An eigenvalue-eigenvector pair satisfies the relation $$A x = \lambda x$$ which has non-trivial solutions if and only if $det(A-\lambda I)=0$.
This is called the characteristic equation, and has $N$ roots, if $A$ is an $N\times N$ matrix. They may be real, complex, simple or multiple roots.
Such systems can be written as $$ \frac{dx}{dt} = P(t) x + g(t) $$ where $P(t)$ is a $2\times 2$ matrix, and $x(t)$ and $g(t)$ are $2\times 1$ column vector. $x_1(t)$ and $x_2(t)$ are called state variables, and the $x_1-x_2$ plane is called the phase plane. A parametric solution curve $(x_1(t),x_2(t))$ is called a trajectory or orbit.
If $g(t)=0$ the system is said to be homogeneous
There are several tools that are very helpful in analyzing behavior
Such systems can be written as $$ \frac{dx}{dt} = [A] x(t) $$ and have solutions of exponential form $x(t)=\vec u e^{\lambda t}$ where $\vec u$ is a vector, and $[A]$ is a constant, square matrix.
Substituting this form into the differential equation, we have $$ \lambda \vec u e^{\lambda t} = [A] \vec u e^{\lambda t} $$ or $$ e^{\lambda t} \{ [A] \vec u - \lambda \vec u \} = 0 $$ which can only happen if $$ [A] \vec u - \lambda \vec u $$ which means that $\vec u$ is an eigenvector of $[A]$ and $\lambda$ is the associated eigenvalue.
The general solution can be written as $$ x(t) = c_1 x_1(t) + c_2 x_2(t) $$ and the initial conditions yield the constraint $$ x_0 = x(0) = c_1 x_1(0) + c_2 x_2(0) $$. This can be written as a matrix equation `[\vec x_1(0)|\vec x_2(0)][[c_1],[c_2]]=x_0`
This has a unique solution as long as the determinant of the matrix `[\vec x_1(0)|\vec x_2(0)]` is non zero. This determinant is called the Wronskiian of the vectors $\vec x_1, \vec x_2$.
For a $2\times 2$ matrix, there are at most two eigenvalues, which can only occur as one of three cases
In this case, the eigenvalues must occur as a conjugate pair $\lambda = \mu \pm i \nu$. Using the identity, $$ e^{\lambda t}=e^{(\mu \pm i \nu)t} =e^{\mu t}e^{\pm i \nu t} = e^{\mu t} (\cos(\nu t) \pm i \sin(\nu t)) $$ the eigenfunctions are $e^{\mu t} \sin(\nu t)$ and $e^{\mu t} \cos(\nu t)$. The resulting phase plane portrain is a periodic orbit (if $\mu=0$), and inward spiral (if $\mu<0$ ) and outward spiral (if $\mu>0$ ).
In this case, if $\lambda_1=\lambda_2=\lambda$ then the two eigenvectors have the t-dependence $e^{\lambda t}$ and $te^{\lambda t}$ respectively. This results in an improper node.
Review table 3.5.1 on page 185, as well as figure 3.5.10 on page 187.
A system of the form $$ \frac{dx}{dt} = f(x(t)) $$ which has no explicit time dependence is called autonomous. Critical points are constant vectors $x_0$ satisfying $f(x_0)=0$.
the critical point is stable if every orbit $\phi(t)$ starting in the neighborhood of the critical point stays in a neighborhood.
$$ \| \phi(0) - x_0 \| < \delta => \| \phi(t) - x_0 \| < \epsilon $$ If $\lim \phi(t)=x_0$ then the critical point is asypmtotically stable.
Nonlinear systems may posess more than one critical point, possibly even an infinite number of critical points (e.g. nonlinear pendulum, page 477).
Curves which separate regions of differing behavior in the phase plane are called separatrices (singular separatrix). In general separatrices connect critical points, and divide the phase plane into disjoint regions.
Given an autonomous system of the form $$ \frac{dx}{dt} = f(x) $$ We can use Taylor's theorem to expand $f(x)$ in the neighborhood of a critical point $x_0$ (where $f(x_0)=0$). We have $$ f(x) = f(x_0) + [J(x)](x-x_0) + ... $$ where $[J]$ is the Jacobian matrix of derivatives `J(x) = [[ \frac{\partial f_1}{\partial x_1}, \frac{\partial f_1}{\partial x_2}],[ \frac{\partial f_2}{\partial x_1}, \frac{\partial f_2}{\partial x_2}]]`
If $\det J(x_0) \neq 0$, then the critical point is isolated. If the jacobian is smooth enough, then the higher order terms are small compared to the linear term, and the phase plane will be a small perturbation of the linear behavior.
Adding a small term $g(t)$ to a linear system $dx/dt = Ax$ (with $det A \neq 0$) causes the critical point to move continuously in the phase plane, and the eigenvalues to vary continuously. Each of the cases remains the same, except for the case of multiple eigenvalues which may split into unequal eigenvalues (see table 7.2.2).
Systems of this type are quadratic, and consist of products of linear terms. $$ dx/dt = x(\epsilon_1 - \sigma_1 x - \alpha_1 y) $$ $$ dy/dt = y(\epsilon_2 - \sigma_2 y - \alpha_2 x) $$ There are at most 4 critical points, and the behavior can be completely determined by calculating the jacobian (and eigenvalues) in terms of the parameters.
These equations are a special case of the competing species equations, with `\sigma_1=\sigma_2=0`. $$ dx/dt = x(\epsilon_1 - \alpha_1 y) $$ $$ dy/dt = y(\epsilon_2 - \alpha_2 x) $$ Two of the critical points vanish, leaving to, one at the origin (0,0) and another generating periodic orbits about the point $(\epsilon_2/\alpha_2,\epsilon_1/\alpha_1)$.
These equations are of the form $y''(t)=f(t,y,y')$ where $f$ is a linear function of $y$, that is $$ y''(t)+p(t)y'(t)+q(t)y(t)=g(t) $$ This can be transformed into a $2\times 2$ system using the substitutions $x_1=y$ and $x_2=y'$.
The following three physical examples all reduce to a second order linear equation:
These are of the form $$ y''(t)+p(t)y'(t)+q(t)y(t)=g(t) $$ If the coefficients $p(t),q(t),g(t)$ are all continuous on an interval time $I=[a,b]$ (containing the initial time $t_0$) then a unique solution of the initial value problem will exist, as long as the coefficients remain continuous. One way to think about this is to state that the solution will continue to exist as long as nothing "goes wrong" with the coefficients in the problem.
Specializing to the homogeneous version of the ode $$ y''(t)+p(t)y'(t)+q(t)y(t)=0 (1) $$ we have the Principle of superposition. If $y_1(t),y_2(t)$ are two solutions of (1), then any linear combination is also $$y_h(t)=c_1 y_1(t) + c_2 y_2(t) $$ This is the general solution of the homogeneous second order equation. If the Wronskian
Specializing to this case $$ y''(t)+by'(t)+cy(t)=0 $$ we observe that solutions must be of the form $y(x)=e^{\lambda t}$.
Substituting this, we conclude that $\lambda$ must satisfy the characteristic polynomial equation $Z(\lambda)=\lambda^2+b \lambda + c=0$. The same values of $\lambda$ result from writing the second order equation as a first order system.
As before, there are 3 cases to consider
For linear constant coefficient differential equations of the form
This has complex roots if the discriminant $b^2-4ac<0$. It will be purely imaginary if, in addition, $b=0$.
To write the solutions in real terms, we use the Euler formula $$exp((a+bi)t)=exp(at)(\cos(bt)+i \sin(bt))$$
We can take the real and imaginary parts as homongenous solutions $$y_1(t)=exp(at)\cos(bt), y_2(t)=exp(at)\sin(bt)$$
Another equation that can have complex roots is the Caucy-Euler equation $$ a t^2 y''(t) + b t y'(t) + c y(t) = 0 $$ This ode has solutions of the form $t^\lambda$. After substituting this into the differential equation, we see that \lambda must satisfy the equation $$ a\lambda(\lambda-1)+b\lambda+c=0 $$ If $\lambda$ is complex, we write $$ t^\lambda = t^{a+ib} = e^{\ln(t)(a+bi)} $$ $$ = e^{a\ln(t)}e^{ib\ln(t)} = t^a (cos(b\ln(t))+i\sin(b\ln(t))) $$
For linear, constant coefficient, second order odes, if the forcing term is polynomial, trigonometric, or exponential (or products thereof), one can find particular solutions of special form.
If $$ a y''(t) + b y'(t) + c y(t) = g(t)$$ then the following cases hold, in general:
NOTE: If the form of the particular solution includes a homogeneous equation, one must multiply for an appropriate power of $t$.
For a second order linear differential equation, or a system of 2 first order linear differential equations, knowledge of the two fundamental (homogeneous) solutions allows one to construct a particular solution.
In the 2x2 matrix case $$ x'(t) = [P(t)] x(t)+g(t) $$ where $x_1(t)$ and $x_2(t)$ are fundamental solutions of $$ x'(t) = [P(t)] x(t) $$ and $[X]=[x_1|x_2]$ is the 2x2 matrix formed from the homogeneous solutions, then the particular solution is formed from $$[X(t)]u(t)$$ where the vector $u(t)$ satisfies the equation $$u'(t)=[X(t)]^{-1}g(t)$$ Integrating this last equation, and multiplying by $[X(t)}$ gives the particular solution.
In the case of a single second order equation $$ y''(t)+p(t)y'(t)+q(t)y(t)=g(t)$$, with the homogeneous solutions $y_1(t)$ and $y_2(t)$, a particular solution will be of the form $$y_p(t)=u_1(t)y_1(t)+u_2(t)y_2(t)$$ where $u_1(t)$ and $u_2(t)$ satisfy the system $$ y_1(t)u_1'(t)+y_2(t)u_2'(t)=0 $$ $$ y_1'(t)u_1'(t)+y_2'(t)u_2'(t)=g(t)$$ Note: The equation must be in standard form, in order to calculate the correct $g(t)$.
The definition of the Laplace Transform is $$F(s)=\int_0^\infty e^{-st}f(t)dt $$
This is well defined for any piecewise continuous functions $f(t)$ which grow at most linearl-exponentially, that is $$|f(t)|\leq Ke^{at}$$ for some $K>0$. The laplace transform is linear. The Laplace transform is a mapping from the physical domain (time) to the transform domain.
The laplace transform of any piecewise continuous, linear-exponentially bounded function, must go to zero as $s \rightarrow \infty$.
The Laplace transform has many properties, the major one being that it maps differential equations in the physical domain to algebraic equations in the transform domain.
If $F(s)=L\{f(t)\}$ then $f(t)=L^{-1}[F(s)]$. $f(t)$ is the inverse laplace transform of $F(s)$. Inverting the laplace transform is equivalent to recognizing which $f(t)$ gives rise to $F(s)$.
Because we cannot pre-compute every possible transform, only the major ones are listed [Table 5.3.1 on page 325 of the text].
The major techniques used to simplfy tranforms are (roughly in order used)
The method of partial fractions is outlined on page 328 of the text.
Completing the square ensures that the constants found in the partial fraction decomposition match the constants appearing in the particular solution.
Solving a linear differential equation by the method of Laplace transforms amounts to performing the following:
If the forcing term is piecewise continuous, or periodic, special methods have been developed. Shifting the forcing term by an amount $t=c$ gives a forcing term of the form $u_c(t)f(t-c)$ where $u_c(t)$ is the Heaviside step function $$ u_c(t)=\{0, 0 < t < c ; 1, t\geq c \}$$ In this case $$L\{u_c(t)f(t-c)\} = e^{cs}F(s) \}$$ Note: it is critical that the values of "c" match in $u_c(t)f(t-c)$.
Generally, in break in smoothness (continuity of function or a derivative) will intruduce a factor $u_c(t)$ in the solution.
Periodic functions: If $f(t+T)=f(t)$ for all $t>0$ then $f(t)$ is periodic, with period $T$. (We assume $T$ is the smallest period).
For such functions, $$L\{f(t)\} = \frac{\int_0^T e^{-st}f(t)dt}{1-e^{-sT}}$$
when solving differential equations with discontinuous forcing functions, the critical first step is to write the forcing term correctly! Some familiarity with infinite (geometric) series is assumed (from Calc II, III).
A unit force, applied over an arbitrarily small time interval, is called an impulse function, \delta(t-t_0). It has the property that
When an impulse function \delta(t-c) is present as a forcing term, the transform of the solution will involve $e^{-cs}$ in the numerator, hence there will be a term u_c(t) present in the solution.
The convolution operator f \star g \equiv \int_0^t f(t-s) g(s) ds is a linear operator. The most important property is
This is a general result for linear constant coefficient differential equations.
Euler's method for the initial value problem
The slope of the tangent to the solution curve at a point t_n is given by m=f(t_n,y(t_n)). Therefore, at the next point t_{n+1}, we have the tangent line approximation y(t_{n+1}) \approx y(t_n)+f(t_n,y(t_n))* (t_{n+1}-t_n). This gives us Euler's method:
II. Integral formulation.
Since any solution of the initial value problem satisfies the identity
A generalization of the tangent line approximating, using polynomials, is called Taylor series. At the point t_n, we have
Euler's method has local truncation error of O(h^2), and a global truncation error of O(h). To improve accuracy, we try to increase the poewr of h. This can be done by using a higher order Taylor series, or a better inegral approximation (see next section).
If we use the trapezoidal method for approximating the integral:
An alternative is to develop a more accurate integrator, called the Runge Kutta method. It uses a sequence of updates to the slopes according to