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Chemistry
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CHEM 5
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Douglas Tobias
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Lecture

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Chemistry

CHEM 5

Douglas Tobias

Fall

Description

LECTURE #8
DIFFERENTIAL EQUATIONS
Differential Equation: the determination of a function that satisfies an equation containing derivatives
of an unknown function.
There are many classes:
1: “Ordinary” differential equations
2 2
Contain only ordinary derivatives dy/dx and d yëdx
2: Partial differential equations
2 2
Contain partial derivaives ▯yê▯x and ▯ y/▯x
The “order” of an ordinary differential equation is the order of the highest derivative that appears in the
equation.
examples:
1: Newton’s second law of motion in one dimension can be written as an ordinary, second order differen-
tial equation
dp/dt = F[t,x(t),p(t)]
where p(t) = mv(t) = mdx/dt is the momentum.
2: The diffusion equation in one-dimension is an example ofa partial differential equation that is second
order in teh spatial variable, x, and first-order in time, t:
2 2
▯cê▯t = D▯ cë▯x
where c is the concentration of the diffusing species, and D is its diffusion constant.
Differential equations are also classified accoring to whetehr they are “linear” or “nonlinear”. The differ-
ential equation
2 2 HnL HnL
f(x, y, dy/dx, d yëdx , .......d yëdx M= 0
2 2 HnL HnL
is said to be linear if f is a linear function of y, dy/dx, d yëdx , .......dyëdx . A general form for a
linear differential equation of order n is:
2 2 HnL HnL
g(x) +a0HxLy + a 1xLdyêdx + a H2Ld yëdx + ....+a HxLdn yëdx = 0
For example:
The two forms of Newton’s second law given above are linear differential equations. The following are
Printed by Wolfram Mathematica Student Edition
examples of nonlinear differential equations:
dy/dx +xy = 0 2 HnL
is said to be linear if f is a linear function of y, dy/dx, d yëdx , .......d Hnyëdx . A general form for a
linear differential equation of order n is:
2 Untitled-2
g(x) +a HxLy + a HxLdyêdx + a HxLd yëdx + ....+a HxLd Hnyëdx HnL= 0
0 1 2 n
For example:
The two forms of Newton’s second law given above are linear differential equations. The following are
examples of nonlinear differential equations:
2
dy/dx +xy = 0
2 2
d yëdx + sinHx + yL= sinx
2 2 2 x
I1 + y Md yëdx + xdyêdx + y = e
Mathematica has several commands for solving a wide variety of differential equations (and systems of
differential equations).
DSolve
Example #1:
dy
Solve the linear, first order, ordinary differential equation, y'= = 3x y, storing the solution in the
dx
variable solution
2
In[1]:=olution = DSolveAy'@xD ã 3 * x * [email protected], [email protected], xE
3
Out[1][email protected] Ø ‰ x [email protected]==
c [1] = arbitrary constant
Example #2:
if we specify initial values for both y and y’ , we get a specific solution that does not contain the arbitrary
constant
say y(0)=1 and y’(0)=0
In[2]:=SolveA9y'@xD ã 3 * x * [email protected], [email protected] ã 1=, [email protected], xE
x3
Out[2][email protected] Ø ‰ ==
Example #3:
solve a first order, nonlinear differential equation, yy’=-x
In[3]:[email protected]@xD * y'@xD ã -x, [email protected], xD
2 2
Out[3]=:[email protected] Ø - -x + 2 [email protected] >, :[email protected] Ø -x + 2 [email protected] >>
Example #4:
Solve a second order, linear differential equation:
In[4]:=olution = [email protected]''@xD + [email protected] ã 0, [email protected], xD
Out[4][email protected] Ø [email protected] [email protected] + [email protected] [email protected]<<
we can use the replacement rule to evaluate the solution for particular values of the arbitrary con-
stants. Here we generate and plot a family of particular solutions
Printed by Wolfram Mathematica Student Edition Untitled-2 3
In[5]:[email protected];
particularSolutions =
[email protected]@xD ê. solution ê. [email protected] Ø i, [email protected] Ø j>
2 2
In[10]:=HermiteH
[email protected], xD gives the Hermite polynomialnH H à.
In[11]:=Hypergeometric1F1
[email protected], b, zD is the Kummer confluent hypergeometric functio1 1 Ha; b; à.
Just like the Solve command, there are many differential equations that cannot be solved exactly using
the DSolve command.
Example #6:
Here is an example of a first-order differential equation that cannot be solved analytically
Printed by Wolfram Mathematica Student Edition 4 Untitled-2
In[12]:[email protected]'@xD ã [email protected] * [email protected], [email protected], xD
Solve::ifun : Inverse functions are being used by Solve, so
some solutions may not be found; use Reduce for complete solution information. à
Solve::ifun : Inverse functions are being used by Solve, so
some solutions may not be found; use Reduce for complete solution information. à
£
Out[12][email protected] @xD ã [email protected] [email protected], [email protected], xD
In this case where exact solutions are not available, numerial solutions can be obtained using NDSolve.
NDSolve
To use NDSolve we need to specify initial conditions (the number of which is at least the order of the
equation) and an interval over which to evaluate the solution:
Example #1:
[email protected]'@xD ã [email protected] * [email protected], [email protected] ã [email protected]<<
The output is given in terms of a strange looking object called an interpolating function, which is a recipe
for representing the solution numerically.
The following shows hwo to translate the interpolating function into a table of values or a plot
In[20]:=lution = [email protected]'@xD ã [email protected] * [email protected], [email protected] ã [email protected],
[email protected] Ø [email protected], 10.<[email protected]<<
we see the interpolating functions again as outputs...
lets store them in the functions f(t) and g(t)
In[28]:=t_D := [email protected] ê. [email protected]@1DD;
[email protected]_D := [email protected] ê. [email protected]@1DD;
[email protected]@tD, [email protected]

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