Solving DE with Constant: Homogenous & Non-Homogenous

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However, this type of DE does not fall under the categories of homogenous or non-homogenous DEs. It is known as a non-homogenous linear DE, and the method for solving it involves finding a particular solution and adding it to the general solution of the associated homogenous equation. In this case, the particular solution is -c/2, and the general solution of the associated homogenous equation y'' + 2y' = 0 is C[1]exp(-2x) + C[2]. Therefore, the final solution is y = -((c x)/2) - 1/2 exp(-2 x) + C[1]exp(-2x) + C[2]. In summary
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phiby
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My math is a little rusty - so please bear with me if this is a stupid question.

I know how to solve both homogenous and non-homogenous DEs

However, I am not sure where a DE with a constant falls & how to solve it.

For eg.

y'' + 2y' + c = 0
(c is a constant).

You cannot convert this into an algebraic equation in r like you do for regular homogenous DEs. So what's the method for solving this?

If this is a different category of DEs (i.e. neither homo nor non-homo), then even giving me the name of this type of DE is good enough - I can google and find the method.

An example of this type of DE is a bar loaded with a uniformly distributed load of f.
The DE is

AEu'' + f = 0

u -> deflection.

f is a constant.
 
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You can solve the equation y'' + 2y' = -c. This is a non homogenous equation where the non-homogenous part is a constant. So the general solution is

-((c x)/2) - 1/2 exp(-2 x) C[1] + C[2]

where C[1] and C[2] are constants.
 

What is a differential equation (DE)?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to describe the relationship between a quantity and how it changes over time or space.

What does it mean for a DE to be constant?

A constant DE is one where the coefficients of the variables and their derivatives do not change with respect to the independent variable. This means that the equation remains the same regardless of the value of the independent variable.

What is a homogenous DE?

A homogenous DE is one where all terms are of the same degree. In other words, the highest derivative term has the same degree as the other terms in the equation.

What is a non-homogenous DE?

A non-homogenous DE is one where the terms are of different degrees. This means that the highest derivative term has a different degree compared to the other terms in the equation.

How do you solve a DE with constant coefficients?

To solve a DE with constant coefficients, you can use the method of undetermined coefficients or the method of variation of parameters. In the method of undetermined coefficients, you assume a solution form and plug it into the DE to find the coefficients. In the method of variation of parameters, you use a variation of the homogeneous solution and solve for a particular solution.

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