gate 2016 ee set 2 - Question 30

Given, \frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}-4\frac{\mathrm{d} y}{\mathrm{d} x}+4y=0

Writing in terms of auxiliary equation: D^{2}y-4Dy+4y=0 \Rightarrow y(D^{2}-4D+4)=0

The roots are D = 2, 2, which are real and equal.

The general equation is: y=(C_{1}+C_{2}x)e^{ax}\Rightarrow (C_{1}+C_{2}x)e^{2x}

Using the given initial conditions to find the values of C_{1} and C_{2}

y(0)=0\Rightarrow x=0, y=0\Rightarrow 0=(C_{1})e^{0}\Rightarrow C_{1}=0

Therefore, y=C_{2}xe^{2x}

y'=C_{2}e^{2x}+2C_{2}xe^{2x}\Rightarrow y'(0)=1, x=0\Rightarrow 1=C_{2}e^{2(0)}+2C_{2}(0)e^{0}\Rightarrow 1=C_{2}

Now the particular solution is: y=xe^{2x} and thus, y(1)=(1)e^{2(1)}=e^{2}=7.38