GATE (TF) Textile 2022 Question Paper Solution | GATE/2022/TF/43

Question 43 (Textile Technology & Fibre Science)

Let $y'$ and $y''$ denote the first and second order derivatives of $y$ with respect to $x$ , respectively. Let $y(x)$ be a solution of the initial value problem:
$y'' - 3y' + 2y = 0$ , $y(0) = 1$ , $y'(0) = 3$ .
Then $y''(0)$ (in integer is) _________.

[Show Answer]

$y'' - 3y' + 2y = 0$ ,

or

$\frac{d^2y}{dx^2} - 3\frac{dy}{dx}+ 2y = 0$ ,

Let
$\frac{dy}{dx}=D$

$\frac{d^2y}{dx^2}=D^2$

$\frac{d^2y}{dx^2} - 3\frac{dy}{dx}+ 2y = 0$ ,

$D^2y-3Dy+2y=0$

$m^2y-3my+2y=0$

$m^2-3m+2=0$

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

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

m=1,2

Solution-

y=Complementry function(C.F.)+Perticular integer(P.I.)

$y=e^2x+e^x$

Given-

y(0)=1
$\frac{dy}{dx}=2e^2x+e^x$

$\frac{d^2y}{dx^2}=4e^2x+e^x$

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