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## Main Question or Discussion Point

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.