Correct answer: 24
\( \lim\limits _{t \rightarrow x} \frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1\)
\(\lim \limits_{\mathrm{t} \rightarrow \mathrm{x}} \frac{2 \mathrm{t} . \mathrm{f}(\mathrm{x})-\mathrm{x}^{2} \mathrm{f}^{\prime}(\mathrm{x})}{1}=1\)
\(2 \mathrm{x} . \mathrm{f}(\mathrm{x})-\mathrm{x}^{2} \mathrm{f}^{\prime}(\mathrm{x})=1\)
\(\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{2}{\mathrm{x}} \cdot \mathrm{y}=\frac{-1}{\mathrm{x}^{2}}\)
\(\text{I.f.}\) \(=\mathrm{e}^{\int-\frac{2}{x} \mathrm{~d} x}=\frac{1}{\mathrm{x}^{2}}\)
\(\therefore \frac{\mathrm{y}}{\mathrm{x}^{2}}=\int-\frac{1}{\mathrm{x}^{4}} \mathrm{dx}+\mathrm{C}\)
\(\frac{\mathrm{y}}{\mathrm{x}^{2}}=\frac{1}{3 \mathrm{x}^{3}}+\mathrm{C}\)
Put \(f(1)=1\)
\(C= \frac23\)
\(y = \frac 1{3x} + \frac{2x^2}3\)
\(y = \frac{2x^3 + 1}{3x}\)
\(f(2) = \frac {17}6\)
\(f(3) = \frac {55}9\)
\(2 f(2) + 3 f(3) = \frac{17}3 + \frac{55} 3 = \frac{72}3 = 24\)
