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

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

Answers (1)

Answer:

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}

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Gurleen Kaur

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