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, $$y(1)=(1)e^{2(1)}=e^{2}=7.38$$