- #1
MAGNIBORO
- 106
- 26
hi, i know a little bit of ODE but not much about PDE,Some math programs give me the solution but I would like to know what methods they use.
The problem is the following:
$$I(a,b) = \int_{0}^{\infty} e^{-ax^{2}-\frac{b}{x^2}}$$
through differentiation under the integral sign, substitution and integration by parts, we can find this properties.
$$I(a,b) = -\sqrt{\frac{b}{a}}\, \left ( \frac{\partial }{\partial b}I(a,b) \right )=-\frac{2a}{1+2\sqrt{ab}} \left ( \frac{\partial }{\partial a} I(a,b)\right )$$
and the condition
$$I(a,0) = \frac{1}{2}\sqrt{\frac{\pi }{a}}$$then using a softfware:
$$I(a,b) = -\sqrt{\frac{b}{a}}\, \left ( \frac{\partial }{\partial b}I(a,b) \right )$$
$$I(a,b) = f(a)\, e^{-2\sqrt{ab}}$$
now with the other equation
$$I(a,b) = -\frac{2a}{1+2\sqrt{ab}} \left ( \frac{\partial }{\partial a} I(a,b)\right )$$
$$I(a,b) = g(b)\, \frac{e^{-2\sqrt{ab}}}{\sqrt{a}}$$
comparing the 2 equations and considering the condition I(a,0) we get
$$I(a,b) = \frac{\sqrt{\pi}}{2} \frac{e^{-2\sqrt{ab}}}{\sqrt{a}}$$To fully understand the development, I would like to know what methods use the program to solve the 2 pde
thanks.
The problem is the following:
$$I(a,b) = \int_{0}^{\infty} e^{-ax^{2}-\frac{b}{x^2}}$$
through differentiation under the integral sign, substitution and integration by parts, we can find this properties.
$$I(a,b) = -\sqrt{\frac{b}{a}}\, \left ( \frac{\partial }{\partial b}I(a,b) \right )=-\frac{2a}{1+2\sqrt{ab}} \left ( \frac{\partial }{\partial a} I(a,b)\right )$$
and the condition
$$I(a,0) = \frac{1}{2}\sqrt{\frac{\pi }{a}}$$then using a softfware:
$$I(a,b) = -\sqrt{\frac{b}{a}}\, \left ( \frac{\partial }{\partial b}I(a,b) \right )$$
$$I(a,b) = f(a)\, e^{-2\sqrt{ab}}$$
now with the other equation
$$I(a,b) = -\frac{2a}{1+2\sqrt{ab}} \left ( \frac{\partial }{\partial a} I(a,b)\right )$$
$$I(a,b) = g(b)\, \frac{e^{-2\sqrt{ab}}}{\sqrt{a}}$$
comparing the 2 equations and considering the condition I(a,0) we get
$$I(a,b) = \frac{\sqrt{\pi}}{2} \frac{e^{-2\sqrt{ab}}}{\sqrt{a}}$$To fully understand the development, I would like to know what methods use the program to solve the 2 pde
thanks.