How can I convert a 2nd order ODE to a 1st order ODE?

In summary: Therefore, if I can find a function u that satisfies those equations, then I can find the y- intercept and solution to the original differential equation.In summary, equation difficulty may be due to derivatives of x and y as well as the function involving t. The standard method would be to define new variables, say z1, z2, z3, and z4 such that z1=x, z2= dx/dt, z3= y, and z4= dy/dt.
  • #1
jrv24
1
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Hi, have this strange 2nd order ODE in one of my tutorials that I am struggling to start. I am not used to dealing with derivatives of both x and y as well as a function involving t.
I was wondering if anyone may be able to point me to the starting line.

I am trying to convert them into 1st order ODEs.

Thanks
 

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  • #2
Do you mean convert those two second order equations to four first order equations? There is no way to reduce a single second order equation to a single first order equation or two second order equations to two first order equations unless you have additional information such as already knowing a solution.

The standard method would be to define new variables, say z1, z2, z3, and z4 such that z1=x, z2= dx/dt, z3= y, and z4= dy/dt. Of course, [itex]d^2x/dt^2= d(dx/dt)dt= dz2/dt[/itex] and [itex]d^2y/dt^2= d(dy/dt)/dt= dz4/dt[/itex].

The equation [itex]d^2x/dt^2+ dy/dt- y+ x= e^t[/itex] becomes [itex]dz2/dt+ z4- z3+ z1= e^t[/itex] or [itex]dz2/dt= -z1+ z3- z4+ e^t[/itex] and [itex]d^2y/dt^2+ 2dx/dt+ y- 2x= 0[/itex] becomes [itex]dz4/dt+ 2z2+ z3- 2z1= 0[/itex] or [itex]dz4/dt= 2z1- 2z2- z3[/itex].

The other two equations are, of course, [itex]dz1/dt= z2[/itex] and [itex]dz3/dt= z4[/itex].
 
  • #3
There is no way to reduce a single second order equation to a single first order equation or two second order equations to two first order equations unless you have additional information such as already knowing a solution.

please , could you re-explain this passage.it is not uderstood for me , thank you
 
  • #4
If you have, say, a second order differential equation and already know one solution you can reduce it to a first order equation for another, linearly independent, solution in much the same way that you can reduce the degree of a polynomial if you already know one of its roots.

For example, I know that one of the roots of the polynomial equation [itex]x^3- 6x^2+ 11x- 6= 0[/itex] is x= 1 and that means that the polynomial has a factor of x- 1. Dividing that polynomial by x- 1 I get \(\displaystyle x^2- 5x+ 6= 0\) as a second degree equation for the other two roots.

Similarly, if I know that [itex]y= e^x[/itex] is a solution to the differential equation y''- 3y'+ 2y= 0, I can "try" a solution of the form [itex]y= u(x)e^x[/itex]. Then [itex]y'= u(x)e^x+ u'(x)e^x[/itex] and [itex]y''=ue^x+ 2u'e^x+ u''e^x[/itex].

Putting those into the
 
  • #5
for reaching out about your struggle with a 2nd order ODE in your tutorial. I understand the frustration that can come with facing new challenges in a field that we are not used to.

To convert a 2nd order ODE to a 1st order ODE, the first step is to introduce a new variable, let's call it u, that is equal to the derivative of y with respect to x. This means that u = dy/dx.

Next, we can rewrite the 2nd order ODE in terms of u and y, rather than x and y. This will result in a system of two 1st order ODEs.

To solve these 1st order ODEs, you can use methods such as separation of variables, substitution, or the method of integrating factors. It is important to note that the initial conditions for the 2nd order ODE will need to be translated into initial conditions for the 1st order ODEs as well.

I hope this helps you get started on converting your 2nd order ODE to a 1st order ODE. Remember to take your time and approach it step by step. If you have any further questions or need clarification, don't hesitate to reach out. Good luck!
 

Related to How can I convert a 2nd order ODE to a 1st order ODE?

1. What is a 2nd order ODE?

A 2nd order ODE (ordinary differential equation) is a mathematical equation that relates a function to its derivatives up to the 2nd order. It is commonly used in physics, engineering, and other scientific fields to describe the behavior of physical systems.

2. What is the difference between a 2nd order ODE and a 1st order ODE?

The main difference between a 2nd order ODE and a 1st order ODE is the number of derivatives involved. A 2nd order ODE involves the first and second derivatives of the function, while a 1st order ODE only involves the first derivative. This means that a 2nd order ODE is more complex and requires more information to solve.

3. Why would you want to convert a 2nd order ODE to a 1st order ODE?

Converting a 2nd order ODE to a 1st order ODE can make the equation easier to solve and can also help in understanding the behavior of the system. It can also make it easier to apply numerical methods or computer simulations to the problem.

4. How do you convert a 2nd order ODE to a 1st order ODE?

To convert a 2nd order ODE to a 1st order ODE, you can introduce a new variable, usually denoted by u, and set it equal to the first derivative of the original function. Then, you can rewrite the 2nd order ODE in terms of u and its derivatives, resulting in a 1st order ODE.

5. What are some applications of solving 2nd order ODEs using 1st order methods?

Solving 2nd order ODEs using 1st order methods can be useful in a variety of fields, such as physics, engineering, and finance. It can be used to model the motion of objects, the behavior of electrical circuits, and the dynamics of chemical reactions. It can also be applied in optimization problems and in analyzing economic systems.

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