Explain solution RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 40 maths

Proved

Hint:

You must know the derivative of exponential function

Given:

$y=3 e^{2 x}+2 e^{3 x}, \text { prove } \frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0$

Solution:

$y=3 e^{2 x}+2 e^{3 x}$

\begin{aligned} &\frac{d y}{d x}=(2)(3) e^{2 x}+(3)(2) e^{3 x} \\ &=6 e^{2 x}+6 e^{3 x} \\ &\frac{d y}{d x}=6 e^{2 x}+\frac{6\left(y-3 e^{2 x}\right)}{2} \\ &\frac{d y}{d x}=6 e^{2 x}+3 y-9 e^{2 x} \\ &=-3 e^{2 x}+3 y \end{aligned}

\begin{aligned} &\text { Again differentiating }\\ &\frac{d^{2} y}{d x^{2}}=\frac{3 d y}{d x}-6 e^{2 x} \quad \ldots \ldots \ldots .(1)\\ &\frac{d y}{d x}-3 y=-3 e^{2 x}\\ &\frac{\frac{d y}{d x}-3 y}{-3}=e^{2 x} \end{aligned}

\begin{aligned} &\text { Put in (1), }\\ &\frac{d^{2} y}{d x^{2}}=\frac{3 d y}{d x}-6\left(\frac{\frac{d y}{d x}-3 y}{-3}\right)\\ &\frac{d^{2} y}{d x^{2}}=\frac{3 d y}{d x}+2 \frac{d y}{d x}-6 y\\ &\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0 \end{aligned}