A number theory problem by Utkarsh Kumar

$15x^2$ - $9$ = $7y^2$

Here $3$ divides LHS so it must also divide RHS.

Let $y$ = $3k$ [ $3$ does not divide $7$ ]

$15x^2$ - $9$ = $63k^2$

$5x^2$ = $3$ + $21k^2$

Here again $3$ divides RHS so it must divide the LHS.

Let $x$ = $3m$

The equation reduces to

$45m^2$ = $21k^2$ + $3$

$15m^2$ = $7k^2$ + $1$

Here $3$ divides LHS but not RHS

Therefore, no natural solution is possible to this equation.