7.7. Ordinary Differential Equations http://www.ck12.org
7.7 Ordinary Differential Equations
General and Particular Solutions
Differential equations appear in almost every area of daily life including science, business, and many others. We
will only considerordinary differential equations(ODE). An ODE is a relation on a functionyof one independent
variablexand the derivatives ofywith respect tox, i.e.y(n)=F(x,y,y′,....,y(n−^1 )). For example,y′′+(y′)^2 +y=x.
An ODE islinearifFcan be written as a linear combination of the derivatives ofy, i.e.y(n)=∑ai(x)y(i)+r(x). A
linear ODE ishomogeneousifr(x) =0.
Ageneral solutionto a linear ODE is a solution containing a number (the order of the ODE) of arbitrary variables
corresponding to the constants of integration. Aparticular solutionis derived from the general solution by setting
the constants to particular values. For example, for linear ODE of second degreey′′+y=0, a general solution has
the formyg=Acosx+BsinxwhereA,Bare real numbers. By settingA=1 andB= 0 ,yp=cosx
It is generally hard to find the solution of differential equations. Graphically and numerical methods are often used.
In some cases, analytical method works, and in the best case,yhas an explicit formula inx.
Multimedia Links
For a video introduction to differential equations(27.0), see Math Video Tutorials by James Sousa, Introduction to
Differential Equations (8:12).
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/612Slope Fields and Isoclines
We now only consider linear ODE of the first degree, i.e. dydx=F(x,y). In general, the solutions of a differential
equation could be visualized before trying an analytic method. Asolution curveis the curve that represents a solution
(in thexy−plane).
Theslope fieldof the differential |eq|uation is the set of all short line segments through each point(x,y)and with
slopeF(x,y).