GATE (TF) Textile 2020 Question Paper Solution | GATE/2020/TF/27

The solution of the differential equation $\frac{d^{2}y}{dx^{2}} + \frac{dy}{dx}-2y =0$ , which satisfies the conditions, $y(0)=0, y'(0)=3$ is___.

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

$\frac{d}{dx}=D$

$(D^2+D-2)y=0$ , D=m

$m^2+m-2=0$

$m^2+2m-m-2=0$

(m+2)(m-1)=0

m=-2, 1

Complementary function(C.F)=y= $c_1 \times e^{x}+c_2 \times e^{-2x}$

Given ,y(0)=0

$0=c_1 \times e^0+c_2 \times e^0$

$c_1=-c_2$ ……………………………….(1)

Given, ${y}'(0)=3$

${y}'=c_1 \times e^{x}-2c_2 \times e^{-2x}$

$3=c_1-2c_2$ ……………………………(2)

By solving equation no. 1 & 2.

$c_1=1 , c_2=-1$

Solution of differential equation, $y=e^{x}-e^{-2x}$ (Ans)