Difference between revisions of "Separation of Variables"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(Added video links)
 
Line 1: Line 1:
−
[https://youtu.be/nzqLplhh0DU Differential Equations: Separation of Variables] by James Sousa
+
== Definition ==
  
−
[https://youtu.be/OF9K9MK0Yvg Ex.1 Differential Equations: Separation of Variables] by James Sousa
+
A ''separable ODE'' is an equation of the form
 +
:<math>x'(t) = g(t) f(x(t))</math>
 +
for some functions <math>g: \mathbb R \to \mathbb R</math>, <math>f: \mathbb R^n \to \mathbb R^n</math>. In this chapter, we shall only be concerned with the case <math>n = 1</math>.
  
−
[https://youtu.be/A_RdFJJUyEg Ex.2 Differential Equations: Separation of Variables] by James Sousa
+
We often write for this ODE
 +
:<math>x' = g(t) f(x)</math>
 +
for short, omitting the argument of <math>x</math>.
  
−
[https://youtu.be/0-dy9aaNdZ8 Ex.3 Differential Equations: Separation of Variables] by James Sousa
+
[Note that the term "separable" comes from the fact that an important class of differential equations has the form
 +
:<math>x' = h(t, x)</math>
 +
for some <math>h: \mathbb R \times \mathbb R^n \to \mathbb R</math>; hence, a separable ODE is one of these equations, where we can "split" the <math>h</math> as <math>h(t, x) = g(t) f(x)</math>.]
  
−
[https://youtu.be/KNQFAusQ_KM Ex 1: Initial Value Problem - Separation of Variables] by James Sousa
+
== Informal derivation of the solution ==
  
−
[https://youtu.be/fBb68MsuAAk Ex 2: Initial Value Problem - Separation of Variables] by James Sousa
+
Using Leibniz' notation for the derivative, we obtain an informal derivation of the solution of separable ODEs, which serves as a good mnemonic.
  
−
[https://youtu.be/nNHlSB6b1HU Solving Separable First Order Differential Equations - Ex 1] by patrickJMT
+
Let a separable ODE
 +
:<math>x' = g(t) f(x)</math>
 +
be given. Using Leibniz notation, it becomes
 +
:<math>\frac{dx}{dt} = g(t) f(x)</math>.
 +
We now formally multiply both sides by <math>dt</math> and divide both sides by <math>f(x)</math> to obtain
 +
:<math>\frac{dx}{f(x)} = g(t) dt</math>.
 +
Integrating this equation yields
 +
:<math>\int \frac{dx}{f(x)} = \int g(t) dt</math>.
 +
Define
 +
:<math>F(x) := \int \frac{dx}{f(x)}</math>;
 +
this shall mean that <math>F</math> is a primitive of <math>\frac{1}{f(x)}</math>. If then <math>F</math> is invertible, we get
 +
:<math>x = F^{-1}\left( \int g(t) dt \right) = F^{-1} \circ G</math>,
 +
where <math>G</math> is a primitive of <math>g</math>; that is, <math>x(s) = F^{-1}(G(s))</math>, now inserting the variable of <math>x</math> back into the notation.
  
−
[https://youtu.be/XExEixAPK6s Solving Separable First Order Differential Equations - Ex 2] by patrickJMT
+
Now the formulae in this derivation don't actually mean anything; it's only a formal derivation. But below, we will prove that it actually yields the right result.
  
−
[https://youtu.be/uS_5bmRUYEI Solving a Separable Differential Equation, Another Example #1] by patrickJMT
+
== General solution ==
  
−
[https://youtu.be/Vc6MwLVdCuM Solving a Separable Differential Equation, Another Example #2] by patrickJMT
+
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 +
:'''Theorem 2.1''':
 +
:Let a separable, one-dimensional ODE
 +
::<math>x'(t) = g(t) f(x(t))</math>
 +
:be given, where <math>f</math> is never zero. Let <math>F</math> be an antiderivative of <math>f</math> and <math>G</math> an antiderivative of <math>g</math>. If <math>F</math> is invertible, the function
 +
::<math>x(t) := F^{-1}(G(t))</math>
 +
:solves the ODE under consideration.
 +
</blockquote>
 +
 
 +
'''Proof''':
 +
 
 +
By the inverse and chain rules,
 +
:<math>\frac{d}{dt} F^{-1}(G(t)) = \frac{1}{\frac{1}{f(F^{-1}(G(t)))}} G'(t) = f(F^{-1}(G(t))) g(t)</math>;
 +
since <math>f</math> is never zero, the fraction occuring above involving <math>f</math> is well-defined.
 +
 
 +
==Resources==
 +
*[https://youtu.be/nzqLplhh0DU Differential Equations: Separation of Variables] by James Sousa
 +
 
 +
*[https://youtu.be/OF9K9MK0Yvg Ex.1 Differential Equations: Separation of Variables] by James Sousa
 +
 
 +
*[https://youtu.be/A_RdFJJUyEg Ex.2 Differential Equations: Separation of Variables] by James Sousa
 +
 
 +
*[https://youtu.be/0-dy9aaNdZ8 Ex.3 Differential Equations: Separation of Variables] by James Sousa
 +
 
 +
*[https://youtu.be/KNQFAusQ_KM Ex 1: Initial Value Problem - Separation of Variables] by James Sousa
 +
 
 +
*[https://youtu.be/fBb68MsuAAk Ex 2: Initial Value Problem - Separation of Variables] by James Sousa
 +
 
 +
*[https://youtu.be/nNHlSB6b1HU Solving Separable First Order Differential Equations - Ex 1] by patrickJMT
 +
 
 +
*[https://youtu.be/XExEixAPK6s Solving Separable First Order Differential Equations - Ex 2] by patrickJMT
 +
 
 +
*[https://youtu.be/uS_5bmRUYEI Solving a Separable Differential Equation, Another Example #1] by patrickJMT
 +
 
 +
*[https://youtu.be/Vc6MwLVdCuM Solving a Separable Differential Equation, Another Example #2] by patrickJMT

Revision as of 01:06, 9 October 2021

Definition

A separable ODE is an equation of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'(t) = g(t) f(x(t))}

for some functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g: \mathbb R \to \mathbb R} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \mathbb R^n \to \mathbb R^n} . In this chapter, we shall only be concerned with the case Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 1} .

We often write for this ODE

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x' = g(t) f(x)}

for short, omitting the argument of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

[Note that the term "separable" comes from the fact that an important class of differential equations has the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x' = h(t, x)}

for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h: \mathbb R \times \mathbb R^n \to \mathbb R} ; hence, a separable ODE is one of these equations, where we can "split" the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(t, x) = g(t) f(x)} .]

Informal derivation of the solution

Using Leibniz' notation for the derivative, we obtain an informal derivation of the solution of separable ODEs, which serves as a good mnemonic.

Let a separable ODE

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x' = g(t) f(x)}

be given. Using Leibniz notation, it becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dt} = g(t) f(x)} .

We now formally multiply both sides by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dt} and divide both sides by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} to obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{f(x)} = g(t) dt} .

Integrating this equation yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \frac{dx}{f(x)} = \int g(t) dt} .

Define

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x) := \int \frac{dx}{f(x)}} ;

this shall mean that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is a primitive of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{f(x)}} . If then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is invertible, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = F^{-1}\left( \int g(t) dt \right) = F^{-1} \circ G} ,

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a primitive of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} ; that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(s) = F^{-1}(G(s))} , now inserting the variable of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} back into the notation.

Now the formulae in this derivation don't actually mean anything; it's only a formal derivation. But below, we will prove that it actually yields the right result.

General solution

Theorem 2.1:
Let a separable, one-dimensional ODE
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'(t) = g(t) f(x(t))}
be given, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is never zero. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} be an antiderivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} an antiderivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is invertible, the function
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t) := F^{-1}(G(t))}
solves the ODE under consideration.

Proof:

By the inverse and chain rules,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dt} F^{-1}(G(t)) = \frac{1}{\frac{1}{f(F^{-1}(G(t)))}} G'(t) = f(F^{-1}(G(t))) g(t)} ;

since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is never zero, the fraction occuring above involving Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is well-defined.

Resources

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Separation_of_Variables&oldid=2020"