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But, a differetial equation is not an equation, right? It’s a definition. Its solution is an equation. Do I understand this correctly?

by nyc111 8y ago

GolDDranks 8y ago
No, it's an equation. Equation is a statement that says that thing A is equal with thing B. The point is that thing A and thing B are often quite dissimilar in form. But since they are the same thing, and because math is deterministic, if you manipulate the both sides the same way (For example, multiply both sides with the same amount), it stays equal. (Plus sometimes you are allowed to manipulate them in different ways, because the axiomatic system you're in says that's allowed for some cases. Example: you are allowed to rewrite (AB)C as A(BC) if you know that the object you are manipulating is associative.) "Solving" an equation means manipulating it in a way that the other side becomes trivial in form. You maybe can't do that for many differential equations, but that doesn't make them less of equations.
- nyc111 8y ago
  > Equation is a statement that says that thing A is equal with thing B. But this describes an equality. I imagine an equation to be a proportionality written with standard units. Aren't they different? I mean an equality, an identity, a definition and an equation are different, even though most of the time they are indicated with the equality sign.
  - creatornator 8y ago
    I believe identities and definitions have different symbols by convention, either the equal-delta sign (definition [0]), or the triple-line equals sign (identity [1]). [0] https://hsm.stackexchange.com/questions/5166/who-first-defined-the-equal-delta-or-delta-over-equal-triangleq-symbol https://hsm.stackexchange.com/questions/5166/who-first-defin... [1] https://www.quora.com/What-does-a-triple-equals-sign-mean https://www.quora.com/What-does-a-triple-equals-sign-mean
- potbelly83 8y ago
  Well technically a PDE is an operator between various function spaces.
randyrand 8y ago
What "solving" means is a bit arbitrary. Solving a differential equations typically means taking one of the sides and going down one order. For example making a new equation 'temperature' = X from 'derivative of temperature' = X

But, a differetial equation is not an equation, right? It’s a definition. Its solution is an equation. Do I understand this correctly?

by nyc111 8y ago

But, a differetial equation is not an equation, right? It’s a definition. Its solution is an equation. Do I understand this correctly?

GolDDranks 8y ago

No, it's an equation. Equation is a statement that says that thing A is equal with thing B. The point is that thing A and thing B are often quite dissimilar in form. But since they are the same thing, and because math is deterministic, if you manipulate the both sides the same way (For example, multiply both sides with the same amount), it stays equal. (Plus sometimes you are allowed to manipulate them in different ways, because the axiomatic system you're in says that's allowed for some cases. Example: you are allowed to rewrite (AB)C as A(BC) if you know that the object you are manipulating is associative.) "Solving" an equation means manipulating it in a way that the other side becomes trivial in form. You maybe can't do that for many differential equations, but that doesn't make them less of equations.

nyc111 8y ago
> Equation is a statement that says that thing A is equal with thing B. But this describes an equality. I imagine an equation to be a proportionality written with standard units. Aren't they different? I mean an equality, an identity, a definition and an equation are different, even though most of the time they are indicated with the equality sign.
- creatornator 8y ago
  I believe identities and definitions have different symbols by convention, either the equal-delta sign (definition [0]), or the triple-line equals sign (identity [1]). [0] https://hsm.stackexchange.com/questions/5166/who-first-defined-the-equal-delta-or-delta-over-equal-triangleq-symbol https://hsm.stackexchange.com/questions/5166/who-first-defin... [1] https://www.quora.com/What-does-a-triple-equals-sign-mean https://www.quora.com/What-does-a-triple-equals-sign-mean
potbelly83 8y ago
Well technically a PDE is an operator between various function spaces.

randyrand 8y ago

What "solving" means is a bit arbitrary. Solving a differential equations typically means taking one of the sides and going down one order. For example making a new equation 'temperature' = X from 'derivative of temperature' = X