Search any question & find its solution
Question:
Answered & Verified by Expert
If \(y=e^{-x} \cos x\) and \(y_4+k y=0\), where \(y_4=\frac{d^4 y}{d x^4}\), then \(k=\)
Options:
Solution:
2439 Upvotes
Verified Answer
The correct answer is:
4
Let \(y=e^{-x} \cos x\)
\(\begin{aligned}
& \mathrm{y}_1=-e^{-x} \sin x-e^{-x} \cos x=-e^{-x} \sin x-y \\
& y_2=-e^{-x} \cos x+e^{-x} \sin x-y_1 \\
& \Rightarrow y_2=-y-y_1+e^{-x} \sin x=-2\left(y+y_1\right) \\
& \Rightarrow y_3=-2\left(y_1+y_2\right)=-2\left(e^{-x} \sin x-y\right) \\
& \Rightarrow y_4=4 y_1+2 y_2=4 y_1-4 y-4 y_1 \text { or } y_4+4 y=0 \\
& \Rightarrow k=4
\end{aligned}\)
\(\begin{aligned}
& \mathrm{y}_1=-e^{-x} \sin x-e^{-x} \cos x=-e^{-x} \sin x-y \\
& y_2=-e^{-x} \cos x+e^{-x} \sin x-y_1 \\
& \Rightarrow y_2=-y-y_1+e^{-x} \sin x=-2\left(y+y_1\right) \\
& \Rightarrow y_3=-2\left(y_1+y_2\right)=-2\left(e^{-x} \sin x-y\right) \\
& \Rightarrow y_4=4 y_1+2 y_2=4 y_1-4 y-4 y_1 \text { or } y_4+4 y=0 \\
& \Rightarrow k=4
\end{aligned}\)
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.