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  • Explain solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple choice Question Question 33 maths textbook solution.

Explain solution for  RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple choice Question Question 33 maths textbook solution.

Answers (1)

Answer : (c)\; k<0

Hint : Use variable separable equation i.e. take the x terms in one side and y terms to the other.

Given : \frac{dy}{dx}-ky=0,y(0)=1 and solution of differential equation approaches to zero when x\rightarrow \infty

Explanation : \frac{dy}{dx}-ky=0

\begin{aligned} &\Rightarrow \frac{d y}{d x}=k y \\ &\Rightarrow \frac{d y}{y}=k d x \end{aligned}

Integrate both sides

\Rightarrow \log y=k x+\log C

Take exponential both sides

\begin{aligned} &\Rightarrow y=C e^{k x} \\ &\text { Now } y(0)=1 \\ &\Rightarrow C=1 \\ &y=e^{k x} \end{aligned}

Now given y \rightarrow 0 \text { when } x \rightarrow \infty \text { if } k=?

\begin{aligned} &\Rightarrow \lim _{x \rightarrow \infty} e^{k x}=0 \quad\left[e^{-\infty}=\frac{1}{\infty}=0\right] \\ &\text { When } k<0 \\ &e^{k x} \rightarrow 0 \text { as } x \rightarrow \infty \end{aligned}

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