# Solving a 2nd order non-linear DE by dimensional analysis/observation

• phantomvommand
In summary, the problem involves solving a 2nd order non-linear DE and the solution involves the equation r = Agt^2. The variable g tilda is included in the equation as a parameter and the constant A must be dimensionless. Dimensional analysis is used to find the functional combination of dimensionless terms in the equation.

#### phantomvommand

Homework Statement
See picture below
Relevant Equations
Dimensional analysis
Solving DE

The top most 2nd order non-linear DE is the one that has to be solved. Below is the solution. This problem is from Morin's Classical Mechanics.
May I know how he could guess that r = Agt^2?
Firstly, why must g tilda be a variable within r? I do not understand what he meant by 'parameter'.
Secondly, how was it deduced that A must be a constant with no units? Aren't there many constants with units?

Delta2
What are the units of g?

Chestermiller said:
What are the units of g?
ms-2?

phantomvommand said:
Homework Statement:: See picture below
Relevant Equations:: Dimensional analysis
Solving DE

View attachment 289987
The top most 2nd order non-linear DE is the one that has to be solved. Below is the solution. This problem is from Morin's Classical Mechanics.
May I know how he could guess that r = Agt^2?
Firstly, why must g tilda be a variable within r? I do not understand what he meant by 'parameter'.
Secondly, how was it deduced that A must be a constant with no units? Aren't there many constants with units?

Have you read up on dimensional analysis?
The notion is that for dimensional consistency any equation can be written as some functional combination of dimensionless terms.

In the present case we believe there is an equation relating ##\tilde g, t, r##, and no other variables of dimension. So it can only involve those variables in dimensionless combinations.
Writing square brackets to mean "dimension of":
##[\tilde g]=LT^{-2}##
##[t]=T##
##[r]=L##
How can these be combined to eliminate the dimensions? Only as ##\frac{\tilde g t^2}r##. So we can write ##A=\frac{\tilde g t^2}r##, where A is dimensionless.

phantomvommand and Chestermiller

## 1. How can dimensional analysis be used to solve a 2nd order non-linear differential equation?

Dimensional analysis involves analyzing the units of each term in the equation to determine the relationship between them. This can help identify any missing terms or constants that may be needed to solve the equation.

## 2. What is the process for solving a 2nd order non-linear differential equation using dimensional analysis?

The first step is to identify the units of each term in the equation. Then, determine the relationship between the units and use this to determine the form of the solution. Finally, use the initial conditions to solve for any unknown constants.

## 3. Can dimensional analysis be used to solve any type of non-linear differential equation?

No, dimensional analysis is most effective for solving equations with simple non-linearities, such as polynomial or exponential functions. It may not be as useful for more complex non-linearities.

## 4. What are the advantages of using dimensional analysis to solve a 2nd order non-linear differential equation?

Dimensional analysis can help simplify the equation and make it easier to understand. It also helps identify any missing terms or constants that may be needed to solve the equation. Additionally, it can provide a more intuitive understanding of the relationship between the variables in the equation.

## 5. Are there any limitations to using dimensional analysis to solve a 2nd order non-linear differential equation?

Yes, dimensional analysis may not be as effective for more complex non-linearities, and it may not provide an exact solution. Additionally, it may not be as useful for equations with multiple variables or terms with non-constant coefficients.