Determine all the pairs of positive odd integers $(a,b),$ such that $a,b>1$ and  $$7\varphi^2(a)-\varphi(ab)+11\varphi^2(b)=2(a^2+b^2),$$ where $\varphi(n)$ is Euler's totient function.
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   If \((a, b)\) is a solution and a prime \(p\) divides both \(a\) and \(b\), then \((ap, bp)\) is also a solution. Since the only prime that divides both \(15\) and \(3\) is \(3\), all solutions to the equation can be expressed in the form \((15 \cdot 3^x, 3 \cdot 3^x)\), where \(x\) is a nonnegative integer.

The final answer is \(\boxed{(15 \cdot 3^x, 3 \cdot 3^x)}\), where \(x\) is a nonnegative integer.