GATE Papers >> Mechanical >> 2017 >> Question No 4

Question No. 4 Mechanical | GATE 2017

The differential equation $\style{font-family:'Times New Roman'}{\frac{d^2y}{dx^2}+16y=0}$ for $\style{font-family:'Times New Roman'}{y\left(x\right)}$ with the two boundary conditions $\style{font-family:'Times New Roman'}{{\left.\frac{dy}{dx}\right|}_{x-0}=1\;\mathrm{and}\;{\left.\frac{dy}{dx}\right|}_{x-\frac{\mathrm\pi}2}=-1}$ has


Answer : (A) no solution


Solution of Question No 4 of GATE 2017 Mechanical Paper

$ \frac{\mathrm d^2\mathrm y}{\mathrm{dx}^2}+16\mathrm y=0\;\;\;\;\frac{\mathrm{dy}}{\mathrm{dx}}\vert_{\mathrm x=0}=0\;\;\;\;\;\;\;\;\;\;\;\frac{\displaystyle\mathrm{dy}}{\displaystyle\mathrm{dx}}\vert_{\mathrm x=\frac{\mathrm\pi}2}=1 $

Auxiliary equation

$ \begin{array}{l}\left(\mathrm D^2+16\right)\mathrm y=0\\3\mathrm D=\pm4\mathrm i\end{array} $

For general solution of differential equation

$ \begin{array}{l}\mathrm y=\left(\mathrm A\;\cos4\mathrm x+\mathrm B\;\sin\;4\;\mathrm x\right)\mathrm e^{0.\mathrm x}\\\mathrm y=-4\mathrm A\;\sin\;4\mathrm x+4\mathrm B\;\cos\;4\mathrm x\end{array} $

Now

$ \mathrm y'\left(0\right)=4\mathrm B=\Rightarrow\mathrm B=\frac14 $

Also

$ \begin{array}{l}\mathrm y'\left(\frac{\mathrm\pi}2\right)=4\mathrm B=-1\\\mathrm B=\frac{-1}4\end{array} $

B is not having value thus no solution

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