Q.4) If $$y=A e^{5 x}+B e^{-5 x}$$, then $$\frac{d^2 y}{d x^2}$$ is equal to
(a) 25 y
(b) 5 y
(c ) -25 y
(d) $$15 y$$

Solution:
Step 1: Find the first derivative of the function
The given function is $$y=A e^{5 x}+B e^{-5 x}$$.
$$
\begin{gathered}
\frac{d y}{d x}=\frac{d}{d x}\left(A e^{5 x}+B e^{-5 x}\right) \\
\frac{d y}{d x}=A \cdot\left(5 e^{5 x}\right)+B \cdot\left(-5 e^{-5 x}\right) \\
\frac{d y}{d x}=5 A e^{5 x}-5 B e^{-5 x}
\end{gathered}
$$

Step 2: Find the second derivative of the function
Differentiate the first derivative result to get the second derivative:
$$
\begin{gathered}
\frac{d^2 y}{d x^2}=\frac{d}{d x}\left(5 A e^{5 x}-5 B e^{-5 x}\right) \\
\frac{d^2 y}{d x^2}=5 A \cdot\left(5 e^{5 x}\right)-5 B \cdot\left(-5 e^{-5 x}\right) \\
\frac{d^2 y}{d x^2}=25 A e^{5 x}+25 B e^{-5 x}
\end{gathered}
$$

Step 3: Relate the second derivative back to the original function
Factor out 25 from the expression:
$$
\frac{d^2 y}{d x^2}=25\left(A e^{5 x}+B e^{-5 x}\right)
$$

Recognize that $$A e^{5 x}+B e^{-5 x}$$ is the original function $$y$$ :
$$
\frac{d^2 y}{d x^2}=25 y
$$

Answer:
The correct option is (a) $$\mathbf{2 5 y}$$.