Interplay between the art of integration and differential equations
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Is there any interplay between "integration theory", if you will, and the theory of differential equations? And if so how exactly is the connection? To be precise, let's take as an example this: $$int f'(g(x))g'(x),dx=f(g(x))+C,$$ this is true since all functions $f,gin mathcal{D}(mathbb{R})$ satisfy the differential equation $f'(g(x))g'(x)=(f(g(x)))'$. The essence of my question is can differential equations be used to find the solutions of integrals.
I know that there's this field called differential Galois theory from which it can be proved that certain functions have no antiderivative in terms of elementary functions, but can it help in the pursuit of solutions of integrals? I also know that the Laplace transform can be used to find integrals which is what prompted my question initially.
integration
asked Jan 6 at 8:31
Stupid Questions IncStupid Questions Inc
7010
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add a comment |
$begingroup$
Is there any interplay between "integration theory", if you will, and the theory of differential equations? And if so how exactly is the connection? To be precise, let's take as an example this: $$int f'(g(x))g'(x),dx=f(g(x))+C,$$ this is true since all functions $f,gin mathcal{D}(mathbb{R})$ satisfy the differential equation $f'(g(x))g'(x)=(f(g(x)))'$. The essence of my question is can differential equations be used to find the solutions of integrals.
I know that there's this field called differential Galois theory from which it can be proved that certain functions have no antiderivative in terms of elementary functions, but can it help in the pursuit of solutions of integrals? I also know that the Laplace transform can be used to find integrals which is what prompted my question initially.
integration
asked Jan 6 at 8:31
Stupid Questions IncStupid Questions Inc
7010
$endgroup$
$begingroup$
I'm not sure if this is what you're looking for, but there is a technique, referred to as differentiation under the integral sign, in which you insert a parameter in an integral, differentiate with respect to that parameter (any number of times), and solve the resulting differential equation. Often, the parameter is inserted in such a way that differentiating with respect to it makes the integral easier to compute.
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– Zachary
Jan 6 at 8:41
1
$begingroup$
@Zachary right, my question is is there some field that puts all of the Feynman trick, laplace transform, ..etc into a coherent structure.
$endgroup$
– Stupid Questions Inc
Jan 6 at 8:42
add a comment |
$begingroup$
Is there any interplay between "integration theory", if you will, and the theory of differential equations? And if so how exactly is the connection? To be precise, let's take as an example this: $$int f'(g(x))g'(x),dx=f(g(x))+C,$$ this is true since all functions $f,gin mathcal{D}(mathbb{R})$ satisfy the differential equation $f'(g(x))g'(x)=(f(g(x)))'$. The essence of my question is can differential equations be used to find the solutions of integrals.
I know that there's this field called differential Galois theory from which it can be proved that certain functions have no antiderivative in terms of elementary functions, but can it help in the pursuit of solutions of integrals? I also know that the Laplace transform can be used to find integrals which is what prompted my question initially.
integration
asked Jan 6 at 8:31
Stupid Questions IncStupid Questions Inc
7010
$endgroup$
Is there any interplay between "integration theory", if you will, and the theory of differential equations? And if so how exactly is the connection? To be precise, let's take as an example this: $$int f'(g(x))g'(x),dx=f(g(x))+C,$$ this is true since all functions $f,gin mathcal{D}(mathbb{R})$ satisfy the differential equation $f'(g(x))g'(x)=(f(g(x)))'$. The essence of my question is can differential equations be used to find the solutions of integrals.
I know that there's this field called differential Galois theory from which it can be proved that certain functions have no antiderivative in terms of elementary functions, but can it help in the pursuit of solutions of integrals? I also know that the Laplace transform can be used to find integrals which is what prompted my question initially.
integration
integration
asked Jan 6 at 8:31
Stupid Questions IncStupid Questions Inc
7010
asked Jan 6 at 8:31
Stupid Questions IncStupid Questions Inc
7010
asked Jan 6 at 8:31
Stupid Questions IncStupid Questions Inc
7010
asked Jan 6 at 8:31
Stupid Questions IncStupid Questions Inc
7010
asked Jan 6 at 8:31
Stupid Questions IncStupid Questions Inc
7010
7010
$begingroup$
I'm not sure if this is what you're looking for, but there is a technique, referred to as differentiation under the integral sign, in which you insert a parameter in an integral, differentiate with respect to that parameter (any number of times), and solve the resulting differential equation. Often, the parameter is inserted in such a way that differentiating with respect to it makes the integral easier to compute.
$endgroup$
– Zachary
Jan 6 at 8:41
1
$begingroup$
@Zachary right, my question is is there some field that puts all of the Feynman trick, laplace transform, ..etc into a coherent structure.
$endgroup$
– Stupid Questions Inc
Jan 6 at 8:42
add a comment |
$begingroup$
I'm not sure if this is what you're looking for, but there is a technique, referred to as differentiation under the integral sign, in which you insert a parameter in an integral, differentiate with respect to that parameter (any number of times), and solve the resulting differential equation. Often, the parameter is inserted in such a way that differentiating with respect to it makes the integral easier to compute.
$endgroup$
– Zachary
Jan 6 at 8:41
1
$begingroup$
@Zachary right, my question is is there some field that puts all of the Feynman trick, laplace transform, ..etc into a coherent structure.
$endgroup$
– Stupid Questions Inc
Jan 6 at 8:42
$begingroup$
I'm not sure if this is what you're looking for, but there is a technique, referred to as differentiation under the integral sign, in which you insert a parameter in an integral, differentiate with respect to that parameter (any number of times), and solve the resulting differential equation. Often, the parameter is inserted in such a way that differentiating with respect to it makes the integral easier to compute.
$endgroup$
– Zachary
Jan 6 at 8:41
$begingroup$
I'm not sure if this is what you're looking for, but there is a technique, referred to as differentiation under the integral sign, in which you insert a parameter in an integral, differentiate with respect to that parameter (any number of times), and solve the resulting differential equation. Often, the parameter is inserted in such a way that differentiating with respect to it makes the integral easier to compute.
$endgroup$
– Zachary
Jan 6 at 8:41
1
1
$begingroup$
@Zachary right, my question is is there some field that puts all of the Feynman trick, laplace transform, ..etc into a coherent structure.
$endgroup$
– Stupid Questions Inc
Jan 6 at 8:42
$begingroup$
@Zachary right, my question is is there some field that puts all of the Feynman trick, laplace transform, ..etc into a coherent structure.
$endgroup$
– Stupid Questions Inc
Jan 6 at 8:42
add a comment |
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$begingroup$
I'm not sure if this is what you're looking for, but there is a technique, referred to as differentiation under the integral sign, in which you insert a parameter in an integral, differentiate with respect to that parameter (any number of times), and solve the resulting differential equation. Often, the parameter is inserted in such a way that differentiating with respect to it makes the integral easier to compute.
$endgroup$
– Zachary
Jan 6 at 8:41
1
$begingroup$
@Zachary right, my question is is there some field that puts all of the Feynman trick, laplace transform, ..etc into a coherent structure.
$endgroup$
– Stupid Questions Inc
Jan 6 at 8:42