let

x = the first number, x>0

y = the second number, y>0

and without lost of generality we can assume x>y

two positive numbers differ by 7

x - y = 7

the sum of their squares is 169

x^2 + y^2 = 169

by solving the system of equations

x - y = 7

x^2 + y^2 = 169

we find that

x = 12

y = 5

we consider only the positive solutions

the numbers are 12 and 5.