Number of integration constants

In summary, the conversation discusses the system of 3 ordinary differential equations in mechanics and how to determine the integration constants when given two initial conditions. It is agreed that one constant can be set to zero, and two integration constants are needed for the general solution. There is also a discussion about the general solution for both velocity and coordinate, and how the initial conditions affect the constants. It is noted that the equation for velocity has three times the derivative of coordinates, which is not Newton's equation of motion.
  • #1
LagrangeEuler
717
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If we have system of 3 ordinary differential equation in mechanics and we have two initial condition ##\vec{r}(t=0)=0## and ##\vec{v}(t=0)=\vec{v}_0 \vec{i}##. If we somehow get
[tex]\frac{d^2v_x}{dt^2}=-\omega^2v_x[/tex]
then [tex]v_x(t)=A\sin(\omega t)+B\cos(\omega t)[/tex]
Two integration constants and one initial condition for velocity. What to do? Should we put that one constant is equal to zero? So ##A=0##, ##B=v_0##?
 
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  • #2
From your solution of velocity,
cooridinate:
[tex]x=A\frac{1-\cos \omega t}{\omega}+B \frac{\sin \omega t}{\omega}[/tex]
accerelation:
[tex]a=A\omega \cos \omega t - B\omega \sin \omega t[/tex]
So
[tex]A= \frac{a_0}{\omega}[/tex]
[tex]B= v_0[/tex]
 
  • #3
Yes, I agree. But suppose that I have only two initial conditions in the beginning. I do not know acceleration. When I integrate ##v_x(t)## I will get
[tex]x(t)=C\cos(\omega t)+D\sin (\omega t)+E[/tex]
so 3 integration constants. And because ##x(t)## satisfy some differential equation of second order, general solution should have only 2 integration constants. And I can put ##D=0##. Is it good reasoning?
 
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  • #4
My formula of x in #2 which includes initial coordinate condition is different from yours. Please criticize it.
 
  • #5
I made a typo. From ##v_x(t)=A\sin \omega t+B \cos \omega t## you can write
##\frac{dx}{dt}=A\sin \omega t+B \cos \omega t##
by integrating
##x(t)=-\frac{A}{\omega}\cos \omega t+\frac{B}{\omega}\sin \omega t+E##.
Of course I can say that ##\frac{A}{\omega}=C## and ##\frac{B}{\omega}=D##
Initial condition is ##x(0)=0##. From that
##0=-\frac{A}{\omega}+E##.
I will get the same result as you.
 
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  • #6
LagrangeEuler said:
anuttarasammyak said:
From your solution of velocity,
cooridinate:
[tex]x=A\frac{1-\cos \omega t}{\omega}+B \frac{\sin \omega t}{\omega}[/tex]
accerelation:
[tex]a=A\omega \cos \omega t - B\omega \sin \omega t[/tex]
So
[tex]A= \frac{a_0}{\omega}[/tex]
[tex]B= v_0[/tex]
I do not think also that this is a general solution. Because by putting ##t=0## first term is always zero, regardless of value of ##A##.
 
  • #7
LagrangeEuler said:
by integrating
sin and cos are to be exchanged.

I agree with your observation that initial coordinate condition does not decide A nor B. As said in #2 initial velocity and initial acceleration decide them.

Popular oscillation equation for x has two constnts to decide the motion, initial cooridinate and initial velocity. Your oscillation equation for v, which has three times derivative of coordinates thus is not Newton's equation of motion, has two constants to decide the motion, initial velocity and initial acceleration. All one rank up in time derivative.
 
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Related to Number of integration constants

What is the meaning of "number of integration constants"?

The number of integration constants refers to the number of arbitrary constants that must be included in the solution of a differential equation in order to account for all possible solutions.

How is the number of integration constants determined?

The number of integration constants is determined by the order of the differential equation. For a first-order differential equation, there is one integration constant, for a second-order differential equation, there are two integration constants, and so on.

What is the significance of the number of integration constants?

The number of integration constants is important because it determines the number of independent solutions to a differential equation. It also allows for the general solution to be expressed in terms of these constants, making it possible to find a specific solution that satisfies initial conditions.

Can the number of integration constants change?

No, the number of integration constants is determined by the order of the differential equation and does not change. However, the values of the integration constants can vary depending on the initial conditions of the problem.

How do integration constants relate to initial conditions?

The values of the integration constants are determined by the initial conditions of the problem. These constants are used to find a specific solution that satisfies the given initial conditions. The number of integration constants must match the number of initial conditions for a unique solution to be found.

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