The correct option is 13 : 17.

Let's find the solution step-by-step.

Step 1: Represent the present ages of the two boys.
The ratio of the present ages of the two boys is given as 3:7.
Therefore, we can represent their current ages as:
Age of the first boy = $3x$
Age of the second boy = $7x$
where $x$ is a common multiplier.

Step 2: Set up the equation using the ratio after 2 years.
After 2 years, their ages will be:
Age of the first boy = $3x+2$
Age of the second boy = $7x+2$

According to the problem, the ratio of their ages after 2 years becomes 5:9. We can write this relation as:
$\frac{3x+2}{7x+2}=\frac{5}{9}$

Step 3: Solve for $x$.
Cross-multiplying both sides of the equation gives:
$9(3x+2)=5(7x+2)$

Expanding the brackets:
$27x+18=35x+10$

Rearranging the terms to solve for $x$:
$18-10=35x-27x$
$8=8x$
$x=1$

Step 4: Find their present ages.
Substitute the value of $x=1$ back into the representations of their present ages:
Present age of the first boy = $3\left(1\right)=3$ years
Present age of the second boy = $7\left(1\right)=7$ years

Step 5: Calculate the ratio of their ages after 10 years.
After 10 years, their ages will be:
Age of the first boy = $3+10=13$ years
Age of the second boy = $7+10=17$ years

Thus, the ratio of their ages after 10 years will be:
$13:17$