Lecture 2 Differential equations

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Lecture 2Differential equations
Physics for informatics
Ing. Jaroslav Jíra, CSc.
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Basic division of differential equations
According to type of derivationOrdinary
Differential Equations ODEThey contain a
function of one independent variable and its
derivatives. Example Partial Differential
Equations - PDE They contain unknown
multivariable functions and their partial
derivatives.Example
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Types of Ordinary Differential Equations
General definition of an ODE
First order differential equationsThe highest
derivative of the function y of independent
variable they contain is one. Example
Higher order differential equations They
contain at least second derivative of the
function yExample
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Types of Ordinary Differential Equations
Linear differential equationsThe function y
appears just linearly here. There are no powers
of the function y and its derivatives, there are
no products of the funciton y and its
derivatives. There are also no functions of the
function y like sin(y), exp(y) etc. Example
Nonlinear differential equations If any of
conditions previously defined for the linear DE
is not met, then we talk about nonlinear
DE.Example
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Types of Ordinary Differential Equations
Homogeneous differential equationsThere is no
constant or function of x on the right side of
the equation. Example Inhomogenous
differential equations Exact opposite to the
homogenous DE.Example
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Types of Ordinary Differential Equations
Differential equations with constant
coefficientsThe function y and all its
derivatives are mutiplied just by constants.
Example Differential equations with
variable coefficients The function y and its
derivatives are mutiplied either by constants or
by functions of x. Example
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How to solve differential equations
The following methods will be mentioned
subsequently Separation of variables
applicable to homogeneous first order ODEs and to
specific types of inhomogeneous first order ODEs.
Characteristic equation applicable to
homogeneous first order or higher order linear
ODEs with constant coefficients. Integrating
factor applicable to inhomogeneous first order
or higher order linear ODEs. This method usually
completes the solution of previous two methods
for inhomogeneous equations. Laplace transform
applicable to homogeneous or inhomogeneous
first order or higher order linear ODEs. This
method transforms differential equation into
algebraic one. Solution of inhomogeneous
differential equations is mostly shorter than the
integrating factor method.
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First order inhomogenous linear differential
equation with constant coefficients separation
of variables
The most simple differential equation
where
Integrating the last equation
Example
Where x2C is general solution of the
differential equation
Sometimes an additional condition is given like

that means the function y(x) must pass through a
point
We have obtained a particular solution y(x)x2
-1
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First order homogeneous linear differential
equation with constant coefficients separation
of variables
The general formula for such equation is
Now we integrate both sides of the equation and
then we apply an exponential function to it
substituting
we obtain general solution
Where C is a constant resulting from the initial
condition
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First order homogenous linear differential
equation with constant coefficients
characteristic equation
The general formula for such equation is
To solve this equation we assume the solution in
the form of exponential function.
If
then
and the equation changes into
after dividing by the e?x we obtain
This is the characteristic equation
the solution is
Where C is a constant resulting from the initial
condition
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Example of the first order linear ODE RC circuit
Find the time dependence of the electric current
i(t) in the given circuit.
Now we take the first derivative of the last
equation with respect to time
characteristic equation is
Constant K can be calculated from initial
conditions. We know that
general solution is
particular solution is
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Solution of the RC circuit in the Mathematica
Given values are R1 kO C100 µF u10 V
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First order inhomogenous linear differential
equation with constant coefficients
integrating factor
The general formula for such equation is
Where P(x) and Q(x) are continuous functions of
variable x.
In the first step we omit the right side taking
Q(x)0 and than we can solve the homogeneous
equation by separation of variables.
Separation of variables gives us
Having solution of homogeneous part of the
equation we can continue by looking for the
integrating factor. Instead of constant C we are
looking for a function C(x), which satisfies both
homogeneous solution and the original
differential equation.
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If C(x) should be a function, then
Substituting for y and y into the original
equation
we obtain
and after small adjustment
Our integrating factor is
The general solution of this inhomogeneous
equation is
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Example of the integrating factor solution LR
circuit
Find the time dependence of the electric current
i(t) in the given circuit.
Firstly we have to solve homogeneous equation
Solution of charact. equation
In the second step we are looking for the
integrating factor
First derivative of the current
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Substituting for i and di/dt into the original
equation gives us
General solution
Knowing that initial current is zero i(0)0, we
can determine K2.
Particular solution of the equation
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Solution of the RC circuit in the Mathematica
Given values are R1 kO L100 mH U10 V
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Second order homogenous linear differential
equation with constant coefficients
The general formula for such equation is
To solve this equation we assume the solution in
the form of exponential function
If
then
and
and the equation will change into
after dividing by the e?x we obtain
We obtained a quadratic characteristic equation.
The roots are
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There exist three types of solutions according to
the discriminant D
1) If Dgt0, the roots ?1, ?2 are real and distinct
2) If D0, the roots are real and identical
?12 ?
3) If Dlt0, the roots are complex conjugated ?1,
?2 where a and ? are real and imaginary parts of
the root
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If we substitute
we obtain
This is general solution in some cases, but
Further substitution is sometimes used
and then
considering formula
we finally obtain
where amplitude A and phase f are constants which
can be obtained from initial conditions and ? is
angular frequency.This example leads to an
oscillatory motion.
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Example of the second order LDE a simple
harmonic oscillator
Evaluate the displacement x(t) of a body of mass
m on a horizontal spring with spring constant k.
There are no passive resistances.
If the body is displaced from its equilibrium
position (x0), it experiences a restoring force
F, proportional to the displacement x
From the second Newtons law of motion we know
Characteristic equation is
We have two complex conjugated roots with no real
part
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The general solution for our symbols is
No real part of ? means a0, and omega in our
case
The final general solution of this example is
Answer the body performs simple harmonic motion
with amplitude A and phase f. We need two initial
conditions for determination of these constants.
These conditions can be for example
From the first condition
From the second condition
The particular solution is
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Example 2 of the second order LDE a damped
harmonic oscillator
The basic theory is the same like in case of the
simple harmonic oscillator, but this time we take
into account also damping.
The damping is represented by the frictional
force Ff, which is proportional to the velocity
v.
The total force acting on the body is
The following substitutions are commonly used
Characteristic equation is
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Solution of the characteristic equation
where d is damping constant and ? is angular
frequency
There are three basic solutions according to the
d and ?.
1) dgt?. Overdamped oscillator. The roots are real
and distinct
2) d?. Critical damping. The roots are real and
identical.
3) dlt?. Underdamped oscillator. The roots are
complex conjugated.
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Damped harmonic oscillator in the Mathematica
Damping constant d1 s-1, angular frequency
?10 s-1
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Damped harmonic oscillator in the Mathematica
All three basic solutions together for ?10 s-1
Overdamped oscillator, d20 s-1Critically
damped oscillator, d10 s-1Underdamped
oscillator, d1 s-1