Differential Equations
A differential equation is an equation that involves a function and its derivatives. Put another way, a differential equation makes a statement connecting the value of a quantity to the rate at which that quantity is changing. For example, for a launching rocket, an equation can be written connecting its velocity to its position, and because velocity is the rate at which position changes, this is a differential equation. Solving this differential equation for the position in terms of time allows the location of the rocket to be determined at any time.
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Examples of Differential Equations
Differential equations frequently appear in a variety of contexts.
Suppose a rocket with mass \(m\) is descending so that it experiences a force of strength \(mg\) due to gravity, and assume that it experiences a drag force proportional to its velocity, of strength \(bv\), for a constant \(b\). Then, Newton's second law tells us that if \(a\) is the acceleration,
\[F=ma=mg-bv\implies \frac{dv}{dt}=g-\frac{b}{m} v,\]
which is a differential equation in \(v\). This is a separable differential equation, and it has solution
\[v=\dfrac{mg}{b}\left(1-e^{-bt/m}\right),\]
and now we can integrate with respect to time to find the position, \(x(t)\), as
\[x(t)=\dfrac{mg}{b}t+\dfrac{m^2g}{b^2}e^{-bt/m}+C,\]
where \(C\) is a constant depending on the initial position \((x_0)\) of the rocket, i.e.
\[C=x_0-\dfrac{m^2g}{b^2}.\]
In biology, we often have a population of \(P\) bacteria that are growing at a rate proportional to the number of bacteria, because the more bacteria there are, the more bacteria are being produced. This means that
\[\frac{dP}{dt}= kP,\]
which is another separable differential equation with solution
\[P=P_0e^{kt},\]
where \(P_0\) is the initial size of the colony.
Many chemical reactions slow down as they proceed because reactants are depleted. For example, the rate at which the concentration of a reactant \(A\), denoted \([A]\), is decreasing is often proportional to the amount of the reactant left, i.e. as there is less reactant, the reaction slows down. This gives the equation
\[\frac{d[A]}{dt}=-k[A],\]
where the negative sign appears because the rate is decreasing. This is another separable equation and has solution
\[[A]=[A]_0e^{-kt},\]
where \([A]_0\) is the initial concentration.
Classifying Differential Equations
To solve differential equations, it's helpful to group them into categories that can be solved with similar techniques, which is done with a variety of words describing them.
An ordinary differential equation is one involving derivatives with respect to only one variable, while a partial differential equation is one involving derivatives with respect to multiple variables. These are often abbreviated as ODE and PDE.
The differential equation \(f'(x)+x^2f(x)=f''(x)\) is ordinary, while the the differential equation \(\frac{\partial f}{\partial x}+\frac{\partial ^2 f}{\partial y^2}=xy\) is a partial differential equation.
The order of a differential equation is the highest-order derivative that appears in it.
The differential equation \(f'(x)+x^2f(x)=f''(x)\) is second-degree because it contains an \(f''(x)\) term, while the the differential equation \(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=xy\) is first-degree, because it only contains first derivatives.
A differential equation is linear if it involves only linear combinations of the derivatives of the unknown function (as opposed to polynomials), and non-linear otherwise.
The differential equation \(f'(x)+x^2f(x)=f''(x)\) is linear because \(f'(x)-f''(x)\) is a linear combination of \(f\) and its derivatives, and the differential equation \(\frac{\partial f}{\partial x}+\frac{\partial ^2 f}{\partial y^2}=xy\) is linear as well. However, an equation like \(f'(x)f(x)+x^2=f''(x)\) is non-linear, because \(f'(x)f(x)\) is a non-linear term.
Common Types of Equations
An arbitrary differential equation is usually very difficult to solve. However, when differential equations take one of several forms, they can be solved exactly:
- Separable differential equations \[\frac{dy}{dx}=f(x)g(y)\]
- First-order linear differential equations \[y'+p(x)y=q(x)\]
- Simple harmonic motion
\[v\frac{dv}{dx} = - \omega^2 x\] - Homogeneous linear differential equations with constant coefficients \[a_ny^{(n)}+\cdots+a_1y'+a_0y=0\]
- Bernoulli's equation \[y'+p(x)y=q(x)y^n\]
- Logistic differential equations \[f'(x) = r\left(1-\frac{f(x)}{K}\right)f(x)\]
- Systems of linear differential equations \[ \begin{align} f_1'(x)&=a_{11}f_1(x)+\cdots+a_{1n}f_n(x)\\ f_2'(x)&=a_{21}f_1(x)+\cdots+a_{2n}f_n(x)\\ &\vdots\\ f_n'(x)&=a_{n1}f_1(x)+\cdots+a_{nn}f_n(x)\\ \end{align} \]
Numerical Solutions
Aside from the forms mentioned above, in most cases, differential equations cannot be solved exactly. The majority of the time, differential equations are solved using numerical approximations, like Euler's method and the Runge-Kutta methods. The solutions are often best understood through computer simulations in these cases, replacing the mathematical problem of solving differential equations with the computational problem of simulating the behavior of a system obeying the equation.
The heat equation in one dimension is \(u_t=\alpha u_{xx}\), where \(u(x,t\)) gives the heat at time \(t\) and position \(x\), and \(\alpha\) is a constant. This equation is relatively intractable, so instead of solving it, we can try to approximate the function \(u(x,t)\) based on the differential equation. This allows us to understand, for instance, the way heat flows through a rod that is heated at one end, even though we can't solve the differential equation involved.
Existence and Uniqueness
Unfortunately, there are some differential equations that can't be solved at all, so one of the most important questions concerning differential equations is which equations we can find solutions to, and when these solutions are unique. A very important result about this is the Picard-Lindelof theorem, which says that differential equations of the form \(y'=F(y)\) have solutions that are unique as long as certain conditions on \(y\) are met. In general, though, determining existence and uniqueness is very hard, so much so that one of the Millennium Prize Problems is concerned with determining the existence and uniqueness of solutions to the Navier-Stokes equations, which are differential equations modeling fluid flow.
References
- Dino, ., & Lfahlberg, . Function_Tangents_Annimation. Retrieved September 7, 2016, from https://en.wikipedia.org/wiki/File:Tangent_function_animation.gif