GATE Papers >> Mechanical >> 2017 >> Question No 127

Question No. 127

Consider the differential equation 3y(x)+27 y (x)=0 with initial conditions y (0)=0 and y’(0)=2000. The value of y at x =1 is___________

Solution of Question No 127 of GATE 2017 Mechanical Paper

$\begin{array}{l}3\mathrm y"+27\mathrm y=0\;\;\;\;\mathrm y(0)=0\;\;\;\;\;\;\;\;\;\;\mathrm y'(0)=2000\\\left(3\mathrm D^2+27\right)\mathrm y=0\\\mathrm D=\pm3\mathrm i\\\mathrm y={\mathrm C}_1\mathrm e^{{\mathrm m}_1\mathrm x}\;+{\mathrm C}_2\mathrm e^{{\mathrm m}_2\mathrm x}\;={\mathrm C}_1\mathrm e^{\left(3\mathrm{ix}\right)}\;+{\mathrm C}_2.\mathrm e^{\left(-3\mathrm{ix}\right)}\\\mathrm y=\mathrm{Asin}3\mathrm x+\mathrm{Bcos}3\mathrm x\\\mathrm y\left(0\right)\Rightarrow0=0+\mathrm{Bcos}\left(0\right)\\\mathrm B=0\\\mathrm y'=3\mathrm A\;\cos3\mathrm x-3\mathrm{Bsinx}\\\mathrm y'\left(0\right)=2000\\2000=3\mathrm A,\\\mathrm A=\frac{2000}3\\\mathrm y=\frac{2000}3\sin3\mathrm x\\\mathrm y\left(\mathrm r=1\right)=\frac{2000}3\sin3=94.08\end{array}$

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