# Differential equation - question

### Differential equation - question

Find a function f(x) which is derivative twice which holds the equation:

-2y'(x) = y(1/x)

have to show a mathematic way and not by guessing
Guest

### Re: Differential equation - question

Guessing (and then checking) is a valid and time honored "mathematical method"!

Now, the great majority of differential equations (in very precise sense, "almost all") have no solution in terms of "elementary functions". What reason do you have to believe this equation does?
Guest

### Re: Differential equation - question

Let z= 1/x. Then x= 1/z= z^{-1} so dx/dz= -z^{-2}. d(y(1/x))/dx= (dy(z))/dz)(dz/dx)= -z^{-2}(dy/dz)= y(z).

dy/y= -z^2 dz

ln(y)= -(1/3)z^3+ C

y(z)= e^{-(1/3)z^3+ C}= C' e^{-(1/3)z^3} where C'= e^C.

So y(x)= C'e^{-1/3x^3}
Guest

### Re: Differential equation - question

Let z= 1/x. Then x= 1/z= z^{-1} so dx/dz= -z^{-2}. d(y(1/x))/dx= (dy(z))/dz)(dz/dx)= -z^{-2}(dy/dz)= y(z).

dy/y= -z^2 dz

ln(y)= -(1/3)z^3+ C

y(z)= e^{-(1/3)z^3+ C}= C' e^{-(1/3)z^3} where C'= e^C.

So y(x)= C'e^{-1/3x^3}
Guest