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solution to integral equation and conic sections

0

$begingroup$

I am trying to understand an integral equation by Euler (English translation of his writings can be found on page 7). A complete background of the problem in physics is problably not necessary, the problem asked is just about mathematics.

Euler considers a force with component functions $X(x)$ and $Y(y)$ and arrives at some point at the equation.

$$ int fracdysqrtB - int Y dy = int fracdxsqrtC- int X dx$$

He then says that when we take functions $X$ and $Y$ which are powers of $x$, we get an equation of the form

$$ int fracdysqrtb^n - y^n = int fracdxsqrta^n - x^n$$

Finally he concludes that

if $n=1$, gives a parabola, $n=2$ an ellipse having centre at C, and yet in this case each integration requires the quadrature of the circle.

• How can I obtain this conclusion mathematically and solve the integral equation? I was able to solve part of the integral, but was unable to deduce the equation of the parabola or ellipse.




• More generally, since it a known result in physics that inverse-square force correspond to orbits which are conic section, shouldn't one find also other solutions (hyperbola)?

calculus integration derivatives physics indefinite-integrals

share|cite|improve this question

edited Mar 26 at 17:03

vromo94

asked Mar 26 at 16:52

vromo94vromo94

12

$endgroup$

add a comment | 

0

$begingroup$

I am trying to understand an integral equation by Euler (English translation of his writings can be found on page 7). A complete background of the problem in physics is problably not necessary, the problem asked is just about mathematics.

Euler considers a force with component functions $X(x)$ and $Y(y)$ and arrives at some point at the equation.

$$ int fracdysqrtB - int Y dy = int fracdxsqrtC- int X dx$$

He then says that when we take functions $X$ and $Y$ which are powers of $x$, we get an equation of the form

$$ int fracdysqrtb^n - y^n = int fracdxsqrta^n - x^n$$

Finally he concludes that

if $n=1$, gives a parabola, $n=2$ an ellipse having centre at C, and yet in this case each integration requires the quadrature of the circle.

• How can I obtain this conclusion mathematically and solve the integral equation? I was able to solve part of the integral, but was unable to deduce the equation of the parabola or ellipse.




• More generally, since it a known result in physics that inverse-square force correspond to orbits which are conic section, shouldn't one find also other solutions (hyperbola)?

calculus integration derivatives physics indefinite-integrals

share|cite|improve this question

edited Mar 26 at 17:03

vromo94

asked Mar 26 at 16:52

vromo94vromo94

12

$endgroup$

add a comment | 

$begingroup$

I am trying to understand an integral equation by Euler (English translation of his writings can be found on page 7). A complete background of the problem in physics is problably not necessary, the problem asked is just about mathematics.

Euler considers a force with component functions $X(x)$ and $Y(y)$ and arrives at some point at the equation.

$$ int fracdysqrtB - int Y dy = int fracdxsqrtC- int X dx$$

He then says that when we take functions $X$ and $Y$ which are powers of $x$, we get an equation of the form

$$ int fracdysqrtb^n - y^n = int fracdxsqrta^n - x^n$$

Finally he concludes that

if $n=1$, gives a parabola, $n=2$ an ellipse having centre at C, and yet in this case each integration requires the quadrature of the circle.

• How can I obtain this conclusion mathematically and solve the integral equation? I was able to solve part of the integral, but was unable to deduce the equation of the parabola or ellipse.




• More generally, since it a known result in physics that inverse-square force correspond to orbits which are conic section, shouldn't one find also other solutions (hyperbola)?

calculus integration derivatives physics indefinite-integrals

share|cite|improve this question

edited Mar 26 at 17:03

vromo94

asked Mar 26 at 16:52

vromo94vromo94

12

$endgroup$

share|cite|improve this question

edited Mar 26 at 17:03

vromo94

asked Mar 26 at 16:52

vromo94vromo94

12

share|cite|improve this question

edited Mar 26 at 17:03

vromo94

asked Mar 26 at 16:52

vromo94vromo94

12

asked Mar 26 at 16:52

vromo94vromo94

12

asked Mar 26 at 16:52

vromo94vromo94

12