1. **State the problem:** Calculate the Wronskian of the functions $y_1 = 5^x$ and $y_2 = 4x^2$, and determine if they are linearly independent or dependent. 2. **Recall the Wronskian formula:** For two functions $y_1$ and $y_2$, the Wronskian $W(y_1,y_2)$ is given by: $$ W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1' $$ 3. **Find the derivatives:** - Derivative of $y_1 = 5^x$ is $y_1' = 5^x \ln(5)$. - Derivative of $y_2 = 4x^2$ is $y_2' = 8x$. 4. **Calculate the Wronskian:** $$ W = 5^x \cdot 8x - 4x^2 \cdot 5^x \ln(5) = 5^x (8x - 4x^2 \ln(5)) $$ 5. **Analyze linear independence:** The Wronskian is not identically zero for all $x$ because $5^x$ is never zero and the expression $8x - 4x^2 \ln(5)$ is not zero for all $x$. Therefore, $W(y_1,y_2) \neq 0$ in general. 6. **Conclusion:** Since the Wronskian is not zero everywhere, the functions $y_1 = 5^x$ and $y_2 = 4x^2$ are linearly independent.