Find simplest differential equation

• epkid08
In summary, to find the simplest differential equation with a given set of particular solutions, you can use the general solution y=C_1y_1+...+C_ny_n and eliminate the constants by successive differentiations. The degree of a differential equation does not necessarily determine its complexity, as there can be multiple ways to form a DE with the same solutions. However, it is important to consider additional solutions that may be introduced when forming a DE.
epkid08
How can I find the simplest differential equation given that I know the set of particular solutions? For example, take $$y_p=\{\ln(x), e^x\}$$, how can I find the simplest differential equation that has these solutions?

Edit: It has to be homogeneous.

If it has to be linear homogeneous, then for example

(-x+x^2\ln(x))y''= (1+x^2\ln(x))y'-(x+1)y

Formal way to find LODE is as follow.
You can form the "general" solution as

y=C_1y_1+...+C_ny_n

and then eliminate C_1,...C_n by successive differentiations.

ex is a solution for y'=y.

ln(x) is a solution for y'=x-1.

Then $$y_p=\{\ln(x), e^x\}$$

satisfy the DE (y'-y)(y'-x-1)= 0

plug the solutions into a general form y''+p*y'+q*y=0 and you get two algebric equations which determines p and q.

Actually how many different ways can we form DE that has solutions y=e^x and y= ln x ?

Is my equation (y'-y)(y'-x-1)= 0 equivalent to (-x+x^2\ln(x))y''= (1+x^2\ln(x))y'-(x+1)y ?

I know a first order nonlinear Riccati DE can be reduced to a linear 2nd order DE. But not particularly sure of this one. Is the degree of a DE has any role to play?
Definition: The degree of a DE is the degree of the highest ordered derivative.

There are uncountable different ways to form DE (it'll be different DEs) that has solutions y=e^x and y= ln x if we do not restrict ourselves by homogeneous linear DEs. For example,

first order ODE
x*(exp(x)-ln(x))*diff(y(x),x)+(-exp(x)*x+1)*y(x)+(-1+x*ln(x))*exp(x)=0 ,

second order ODE
-ln(x)*x^2*(ln(ln(x))*ln(x)-1)*diff(y(x),x)^2-y(x)*x*(ln(x)+1)*diff(y(x),x)+y(x)*ln(x)*x^2*(ln(ln(x))*ln(x)-1)*diff(y(x),\$(x,2))+factor(y(x)^2*ln(y(x))*ln(x)+y(x)^2*ln(y(x)))=0 .

Even just looking at the form (y' - y)(y' - x-1) = 0, there are any number of ways to form a differential equation with the given solutions, since Dn(ex) = ex, one can use (y(n) - y)(y' - x-1) = 0, for any positive integer n.

Last edited:
pbandjay said:
Even just looking at the form (y' - y)(y' - x-1) = 0, there are any number of ways to form a differential equation with the given solutions, since Dn(ex) = ex, one can use (y(n) - y)(y' - x-1) = 0, for any positive integer n.

You have to be careful, though. You proposed DE has introduced new solutions: $\exp[\omega_n x]$, where $\omega_n$ is an n-th root of 1. So while such a DE does have the desired solutions, it now has many more, which may not be desired.

OK I will take note that 'simplest DE' means linear DE not the lowest order DE.

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model a wide range of phenomena in science and engineering, including motion, growth, and change over time.

2. Why is it important to find the simplest form of a differential equation?

Finding the simplest form of a differential equation can make it easier to understand and solve. It also allows for more efficient and accurate calculations, which is crucial in many scientific applications.

3. How do you determine the simplicity of a differential equation?

The simplicity of a differential equation is determined by the number of terms and operations present in the equation. The fewer terms and operations, the simpler the equation is considered to be.

4. What are the benefits of using a simple differential equation in scientific research?

Using a simple differential equation can help scientists make predictions and better understand the behavior of a system. It also allows for easier integration with other mathematical models and theories.

5. Can all differential equations be simplified into a simpler form?

No, not all differential equations can be simplified into a simpler form. Some equations are inherently complex and cannot be reduced. However, scientists often aim to simplify equations as much as possible to make them more manageable and easier to work with.

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