1.1 Definitions

1.1 Definitions

DE:  An equation with derivatives in it

ODE: A DE with only one independent variable.

PDE: A DE with more than one independent variable.

Order of a DE:  The highest derivative appearing in a DE

linear DE:  the dependent variable variable (and derivatives) only turn up as linear combinations.

Normal Form of a DE is solved for the highest derivatives.

Solution of a DE: A function that satisfies the DE on some set

Explicit Solution of a DE: An explicit function that satisfies the DE on some set.

Implicit Solution of a DE: An implicit function that satisfies the DE on some set.

Families of Solutions of a DE: A bunch of solutions that depend on some parameter(s) e.g. constant of integration.

System of DE: A bunch of DE for a bunch of functions

1.1 Examples

Ex1

A second order ODE with a 2 parameter familly of solutions defined for all real t.

DE1 =  y''[t] +9 y[t] == Sin[t]
Simplify[DSolve[ DE1,y[t],t]]

Ex2

A second order PDE with boundary conditions and intial conditions

DE2 =  (∂_(x,x))u[x,t]-(∂_(t,t))u[x,t]==0.1 Sin[x]
{L, TMax}={2 π, 12};
uSol[x_,t_]=u[x,t]/.NDSolve[{
DE2,
u[0,t]==u[L,t],
u[x,0]==Cos[x],
Derivative[0,1][u][x,0]==0
},
u[x,t],
{x,0,L},
{t,0, TMax}][[1]]
Manipulate[
Plot[
uSol[x,t],
{x, 0, L},
PlotRange->{-1,1}
],
{t,0, TMax}
]

Ex3

A system of 2 1st order ODE with initial conditions.  (predator-prey p107-108)

DE3 = {
x'[t] == x[t] ( -a + b y[t]),
y'[t]== y[t](d - c x[t])
}

TMax = 32;
{a,b,c,d,x0,y0}=RandomReal[{0.3,4},6]
{xSol[t_],ySol[t_]}={x[t],y[t]}/.NDSolve[
 {
x'[t] == x[t] ( -a + b y[t]),
y'[t]== y[t](d - c x[t]),
x[0]==x0,
y[0]==y0
},
{x[t],y[t]},
{t, 0,TMax}
][[1]];
LogPlot[{xSol[t],ySol[t]},{t,0, TMax},
PlotRange->All,
Frame->True,
Axes->False,
FrameLabel->{"t","x and y"},
PlotStyle->{Red, Blue}]

ParametricPlot[{xSol[t],ySol[t]},{t,0, TMax},
PlotRange->All,
Frame->True,
Axes->False,
FrameLabel->{"x","y"},
AspectRatio->GoldenRatio]

Ex4

Verifyng that something is a solution.  Here is a second order ODE with a two parameter familly of solutions. Suppose we do not trust the software how can we check?

DE4 = y''[t] -1/t  y'[t]==1
DSolve[DE4,y[t],t]

Lets look at the bit without any constants of integration

ySol[t_]= -((t^2)/4)+(1/2) (t^2) Log[t]

ySol''[t]

1/t ySol'[t]

Looks like it works but what is the domain?

Lets look at the bit with constants of integration

yc[t_]=1
1/t yc'[t]
yc''[t]

yc[t_]= (t^2)/2
1/t yc'[t]
yc''[t]

Ex5

Verify that  (x^3)+3 (y^2)==C is an implicit solution of 6 y dy/dx+3(x^2)==0 for any constant C.

D[ (x^3)+3 ((y[x])^2)==c,x]