GATE Papers >> Mechanical >> 2018 >> Question No 146

Question No. 146 Mechanical | GATE 2018

Given the ordinary differential equation

                     $ \frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0 $ 

with $ y\left(0\right)=0 $ and $ \frac{dy}{dx}\left(0\right)=1 $ , the value of $ y\left(1\right) $ is _________ (correct to two decimal places).


Answer : 1.45 to 1.48


Solution of Question No 146 of GATE 2018 Mechanical Paper

$ \frac{\mathrm d^2\mathrm y}{\mathrm{dx}^2}+\frac{\mathrm{dy}}{\mathrm{dx}}-6\mathrm y=0 $

The auxiliary equation is : $ \left(\mathrm D^2+\mathrm D-6\right)y=0 $

$ \begin{array}{l}\left(\mathrm D+3\right)\left(\mathrm D-2\right)=0\\\therefore\mathrm D=-3,2\end{array} $

The solution is $ \mathrm y={\mathrm C}_1\mathrm e^{-3\mathrm x}+{\mathrm C}_2\mathrm e^{2\mathrm x} $

$ \therefore\frac{\mathrm{dy}}{\mathrm{dx}}=-3{\mathrm C}_1\mathrm e^{-3\mathrm x}+2{\mathrm C}_2\mathrm e^{2\mathrm x} $

Given that,

When $ \mathrm x=0,\;\mathrm y=0 $

$ 0={\mathrm C}_1+{\mathrm C}_2.............(\mathrm i) $

Also, when $ \mathrm x=0\;;\;\frac{\mathrm{dy}}{\mathrm{dx}}=1 $

$ -3{\mathrm C}_1+{\mathrm C}_2=1...............(\mathrm{ii}) $

From (i) & (ii)

$ \begin{array}{l}{\mathrm C}_2=\frac15,\;{\mathrm C}_1=-\frac15\\\mathrm y=\frac15\mathrm e^{2\mathrm x}-\frac15\mathrm e^{-3\mathrm x}\\\mathrm{When}\;\mathrm x=1\\\mathrm y\left(1\right)=\frac15\mathrm e^2-\frac15\mathrm e^{-3}\\\mathrm y\left(1\right)=\frac15\left(\mathrm e^2-\mathrm e^{-3}\right)=\frac{7.34}5=1.468\end{array} $

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