1. Lecture 2 Differential equations
Physics for informatics
Lecture 2 Differential equations
Ing. Jaroslav Jíra, CSc.

2. Differential equations
The most simple differential equation:
where
We are looking for the function
Solution of such equation is
Example:
Where x2+C 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

3. First 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 the equation will change into
after dividing by the eλx we obtain
aλ+b is the characteristic equation
the solution is
Where C is a constant resulting from the initial condition

4. Example of the first order LDE – RC circuit
Find the time dependence of the electric current i(t) in the given circuit.
Now we take the first derivative of the right 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

5. Solution of the RC circuit in the Mathematica
Given values are R=1 kΩ; C=100 μF; u=10 V

6. The general formula for such equation is
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 eλx we obtain
We obtained a quadratic characteristic equation. The roots are

7. There exist three solutions according to the discriminant D
1) If D>0, the roots λ1, λ2 are real and different
2) If D=0, the roots are real and identical λ12 =λ
3) If D<0, the roots are complex conjugate λ1, λ2 where α and ω are real and imaginary parts of the root

8. If we substitute
we obtain
This is the solution in some cases, but …
Further substitution is sometimes used
and then
considering formula
we finally obtain
where amplitude A and phase φ are constants which can be obtained from the initial conditions and ω is angular frequency. This example leads to an oscillatory motion.

9. 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 (x=0), 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 conjugate roots with no real part

10. The general solution for our symbols is
No real part of λ means α=0, 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 φ. 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

11. 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

12. Solution of the characteristic equation
where δ is damping constant and ω is angular frequency
There are three basic solutions according to the δ and ω.
1) δ>ω. Overdamped oscillator. The roots are real and different
2) δ=ω. Critical damping. The roots are real and identical.
3) δ<ω. Underdamped oscillator. The roots are complex conjugate.

13. Damped harmonic oscillator in the Mathematica
Damping constant δ=1 [s-1], angular frequency ω=10 [s-1]

14. Damped harmonic oscillator in the Mathematica
All three basic solutions together for ω=10 s-1 Overdamped oscillator, δ=20 s-1 Critically damped oscillator, δ=10 s-1 Underdamped oscillator, δ=1 s-1