Separation of variables , but for 2nd order

In summary, the conversation discusses a problem involving separation of variables in solving a second order ODE. The problem requires reduction of order, resulting in u''*sinx-2u'*cosx=0. The process of solving this problem involves turning it into u''/u'=-2cosx/sinx and using substitution techniques such as u substitution or w substitution. The conversation ends with the understanding that the problem is not as complicated as originally thought.
  • #1
ericm1234
73
2
"separation of variables", but for 2nd order

Ok, I know how to separate variables in solving an ODE. I am unable to understand a solution I have for a problem which was the result of reduction of order- we end up with u''*sinx-2u'*cosx=0
so turn this into u''/u'=-2cosx/sinx
At this point I don't understand the process. I've tried looking at it as making the left side a "u substitution", or rewriting it d(du/dt)/dt / du/dt and integrating both sides,
also, tried making a substition w=u', but this seems hard to get u back in the end.
Can someone explain thoroughly what the process looks like in solving this?
 
Physics news on Phys.org
  • #2
hi ericm1234! :wink:
ericm1234 said:
u''/u'=-2cosx/sinx
…
also, tried making a substition w=u', but this seems hard to get u back in the end.

well, then that's w'/w = -2cosx/sinx,

so ln(w) = … ? :smile:​
 
  • #3


Thanks, got it now, not as complicated as I was making it.
 

1. What is the concept of separation of variables for 2nd order equations?

The concept of separation of variables for 2nd order equations is a technique used to solve differential equations with two independent variables. It involves breaking down the equation into two separate equations, each with one independent variable, and then finding solutions to each equation individually.

2. How is the process of separation of variables for 2nd order equations different from that of 1st order equations?

The process of separation of variables for 2nd order equations is different from that of 1st order equations because it requires the use of partial derivatives, as opposed to ordinary derivatives. This is due to the presence of two independent variables in the equation.

3. What types of equations can be solved using separation of variables for 2nd order equations?

Separation of variables for 2nd order equations is typically used to solve partial differential equations, which involve multiple independent variables. This method is also applicable to certain ordinary differential equations with two independent variables.

4. Are there any limitations to using separation of variables for 2nd order equations?

One limitation of separation of variables for 2nd order equations is that it can only be applied to equations that can be separated into two equations with one independent variable each. Additionally, this method may not always provide a complete solution to the equation and may require additional techniques to find a general solution.

5. How is separation of variables for 2nd order equations useful in scientific research?

Separation of variables for 2nd order equations is a powerful tool in mathematical modeling and scientific research. It allows scientists to solve complex differential equations and analyze the behavior of systems with multiple variables. This method is particularly useful in fields such as physics, engineering, and economics.

Similar threads

MHB Solving Rational ODE's of the form (ax+by+c) dx+(ex+fy+g) dy=0.
    • Differential Equations
Replies
1
Views
5K
MHB Change variable to solve equation
    • Differential Equations
Replies
14
Views
2K
MHB Separation of Variables/ODE
    • Differential Equations
Replies
4
Views
2K
MHB How Do I Solve this 2nd Order Non-Linear ODE and Find Its Equilibrium Points?
    • Differential Equations
Replies
3
Views
2K
I Change of variable (2nd order ODE)
    • Differential Equations
Replies
8
Views
2K
I Understanding the change of variables for PDEs
    • Differential Equations
Replies
2
Views
2K
I Can you check the solution for this second order ODE?
    • Differential Equations
Replies
6
Views
2K
MHB Matrix Methods for Difference Problems
    • Differential Equations
Replies
3
Views
2K
Solving a Non-Homogeneous Linear Differential Equation
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
Solving a separable matrix ODE.
    • Calculus and Beyond Homework Help
Replies
3
Views
311

Hot Threads

Recent Insights

Back
Top