a. Hint: Let
b. Hint:OrdinaryDifferentialEquations
Generaland ParticularSolutions
Differentialequationsappearin almosteveryarea of daily life includingscience,business,and manyothers.
We will only considerordinarydifferentialequations(ODE).An ODEis a relationon a functionyof one in-
dependentvariablexand the derivativesofywith respecttox, i.e.y(n)=F(x,y,y',....,y(n- 1)). For example,
y' + (y' )^2 + y = x.
An ODE is linear if F can be written as a linear combination of the derivatives of y, i.e.
. A linearODEishomogeneousifr(x) = 0.
Ageneralsolutionto a linearODEis a solutioncontaininga number(the orderof the ODE)of arbitrary
variablescorrespondingto the constantsof integration.Aparticularsolutionis derivedfrom the generalso-
lutionby settingthe constantsto particularvalues.For example,for linearODEof seconddegreey' + y =
0, a generalsolutionhavethe formyg= A cos x + B sin x whereA, B are real numbers.By setting
A = 1 and B = 0, yp= cos x
It is generallyhard to find the solutionof differentialequations.Graphicallyand numericalmethodsare often
used.In somecases,analyticalmethodworks,and in the best case,yhas an explicitformulainx.
SlopeFieldsand Isoclines
We now only considerlinearODEof the first degree,i.e.. In general,the solutionsof a dif-
ferentialequationcouldbe visualizedbeforetryingan analyticmethod.Asolutioncurveis the curvethat
representsa solution(in thexy- plane).
Theslopefieldof the differential|eq|uationis the set of all shortline segmentsthrougheachpoint(x,y) and
with slopeF(x,y).