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

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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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