Linearity / multilinearity in LDE(linear differential equations)

[shivaniits]

hello everyone
we demonstrate the linearity in a function by a superposition principle..as in f(x)=y
f(x1+x2)=f(x1)+f(x2)
but that' the case when we have a single variable as x and if we have two variables then we modify the concept of linearity to multilinearity where f(x,y)=z
can never be represented as: f(x1+x2,y1+y2)=f(x1,y1)+f(x2,y2)
but if we have to apply the concept of multilinearity in LDE(linear differential equations)..?
well you will say what an idiot..i mean we have rules to check if differential equation is linear or not(well my book says that)
the rules are: the differential equations are those in which the dependent variable and its derivative occur only in first degree and are not multiplied together
i tried to match this check for linearity against the one which we have been taught while solving equations..the superposition principle..
let me explain about what i am trying to ask with this super slow typing speed
please let me what do you think and do correct me if i am wrong in considering all this..!

 

[jvicens]

Hello, thank you for bringing up this topic. It is true that the concept of linearity in a function can be extended to multilinearity when dealing with multiple variables. However, when it comes to linear differential equations (LDEs), the concept of linearity is slightly different.

In LDEs, we are dealing with a dependent variable and its derivatives, and the linearity refers to the relationship between the dependent variable and its derivatives. The rules you mentioned are correct – in a linear differential equation, the dependent variable and its derivatives should only occur in the first degree and should not be multiplied together.

This is different from the superposition principle, which deals with the linearity of the function itself. In LDEs, we are looking at the linearity of the relationship between the dependent variable and its derivatives, not the linearity of the function itself.

So, while the superposition principle may not directly apply to LDEs, the concept of linearity is still important in determining whether an equation is linear or not. I hope this clarifies your doubts. Please let me know if you have any other questions or if I can provide further clarification.