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Introduction to Differential Equations

April 5th, 2009 |

Author: Merrin

Differential equations are much more juicier than the bland equations we have encountered until now: linear equations, quadratic equations, and systems of linear equations for example. We have learned so far how to write limits, derivatives, and integrals which gave us much more exciting equations, but still most of these equations were merely just identifications of the new functions.

$\cfrac{dy}{dx} =f(x)\qquad \displaystyle\int f(x)dx=F(x)$

Differential equations contain derivative terms of various order. Notice that with equations up until now we typically solve for the “points” where an equation is valid f(x)=0, for example. In a differential equation, the solution is a “function” that satisfies the equation. This is quite a different situation.

Perhaps the simplest differential equation which we should be able to find at least one solution is

$\displaystyle\cfrac{dy}{dx} =y$

What is this equation is asking for? Find the function y such that its derivative is itself. We know the one function which has that property, the exponential of x. The solution by guessing is

$\displaystyle y=e^{x}$

We can always test a solution of a differential equation by plugging it into the equation.

$\displaystyle \cfrac{d}{dx} e^{x} =e^{x}$

But not so fast, is this really the only solution to the differential equation. Recall when we went to find antiderivatives we could add an arbitrary constant. It doesn’t seem to be the case here that we can add a constant to our solution since the derivative of a constant is zero then both sides wouldn’t match.

$\displaystyle \cfrac{d}{dx} (e^{x} +c)\ne (e^{x} +c)$

But if we multiply our solution by a constant, we get a broader family of solutions.

$\displaystyle \cfrac{d}{dx} (ce^{x} )=ce^{x}$

A general solution to a differential equation may have a family of solutions parameterized by constants. We can find the value of the constants for a particular solution if we are given some specific information. Suppose it is given that

$y'(0)=e$

This is called an initial value problem. Now we can take our general solution and find a specific solution

$\displaystyle y'(0)=ce^{0}$
$c=e$
$\displaystyle y(x)=e^{x+1}$

There are some direct methods to solve differential equations besides just guessing the answer in your head. For the example I gave, the $x$ and $y$ variables can be separated to either side of the equal sign, and then integrated.

$dy=ydx$
$\displaystyle \cfrac{dy}{y} =dx$

Now we take the indefinite integrals of both sides of the equation and combine the integration constants into C.

$\displaystyle \int \cfrac{dy}{y} =\int dx$
$\displaystyle \ln \left|y\right|=x+C$
$\displaystyle e^{\ln \left|y\right|} =e^{x+C} =c_{1} e^{x}$
$\displaystyle \left|y\right|=c_1e^{x}$
$\displaystyle y=\pm c_1e^{x}=Ae^{x}$

Equations solvable by this technique are called separable. We will investigate several other types of differential equations in this chapter that are carefully controlled to give simple solutions.

On the other side of the coin we can write equations in terms of relations between integrals. These equations are called integral equations and are much harder to solve than differential equations in general. Here is an example of an integral equation.

$\displaystyle \int _{a}^{x}(x-t)y(t) dt=g(x)$

With the goal of solving for y(x), we take the derivative of both sides of the equation. We know how to differentiate an integral We use Leibnitz formula.

$\displaystyle \cfrac{d}{dx}\int_{u(x)}^{v(x)}f(t,x)dt=v'f(v,x)-u'f(u,x)+\int_{u(x)}^{v(x)}\cfrac{\partial }{\partial x} f(t,x)dt$
$\displaystyle \cfrac{d}{dx} \int _{a}^{x}(x-t)y(t)dt =g'(x)$
$\displaystyle {(1)(x-x)y(x)-(0)+\int _{a}^{x}y(t)dt =g'(x)}$
$\displaystyle \int _{a}^{x}y(t)dt =g'(x)$

Now we take one more derivative and we have

$\displaystyle \cfrac{d}{dx} \int _{a}^{x}y(t)dt =g''(x)$
$\displaystyle (1)y(x)-(0)=g''(x)$
$\displaystyle y(x)=g''(x)$

This is the solution of one of the most basic integral equations. Integral equations are generally considered more advanced and won’t be covered in this book. Try to learn integral equations after differential equations.

In this chapter, we will learn how solve a set of well behaved differential equations that have elementary solutions. These equations will all be ordinary differential equations. This nomenclature we describe below.

Differential equations are equations that have derivatives as terms or terms containing derivatives.

$\displaystyle m\cfrac{d^{2} x}{dt^{2} } =F(t)$
$\displaystyle \arcsin \left(\cfrac{d^{n} y}{dx^{n} } \right)=\cos y$

Ordinary differential equations don’t contain partial derivatives just regular derivatives of a single function usually taken as x in mathematics or as other physical quantities in science or engineering applications.
The order of a differential equation is the order of the highest derivative in the equation. This equation is of order n.

$\displaystyle \cfrac{d^{n} y}{dx^{n} } +y^{2n} =a$

Linear differential equations of second order look like this

$\displaystyle a(x)\cfrac{d^{2} y}{dx^{2} } +b(x)\cfrac{dy}{dx} +c(x)y=q(x)$

Constant coefficient differential equations for second order look like this

$\displaystyle a\cfrac{d^{2} y}{dx^{2} } +b\cfrac{dy}{dx} +cy=q(x)$

When q(x) is zero the previous two equations are said to be homogeneous, otherwise they are called nonhomogeneous or inhomogeneous. Nonlinear differential equations contain derivatives but cannot be cast in the general linear form.

$\displaystyle {\cfrac{dy}{dx} +\sin y=q(x)}$
$\displaystyle {\sqrt{\cfrac{dy}{dx} } _{}^{} +y=\cos x}$

We will solve a selection of equations that have elementary methods of solutions.
First order separable equations (can be nonlinear)

$\displaystyle \cfrac{dy}{dx} =f(x)g(y)$

Exact differential equations are a certain class of equations with conditions on M and N of the form

$\displaystyle M(x,y)+N(x,y)\cfrac{dy}{dx} =0$

First order linear nonhomogeneous equations

$\displaystyle \cfrac{dy}{dx} +p(x)y=q(x)$

Second order homogeneous linear constant coefficient equations

$\displaystyle a\cfrac{d^{2} y}{dx^{2} } +b\cfrac{dy}{dx} +cy=0$

Second order nonhomogeneous linear constant coefficient equations

$\displaystyle a\cfrac{d^{2} y}{dx^{2}} +b\cfrac{dy}{dx} +cy=q(x)$

We will restrict our studies to where q(x) is a select product or sum of possible functions such as polynomials, sines or cosines, and exponentials.

Second order Euler equations

$\displaystyle ax^{2} \cfrac{d^{2} y}{dx^{2} } +bx\cfrac{dy}{dx} +cy=0$

One of the most important application of differential equations in physics is Newton’s first law which states that

$m\cfrac{d^{2} x}{dt} = F(t,x)$

A small investment of time in terms of learning these introductory differential equations can help you begin to understand many actual applications of calculus such as radiocarbon dating, chemical reaction theory, exponential growth, poulation dynamics, mechanics, and much more.

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