GATE (TF) Textile 2021 Question Paper Solution | GATE/2021/TF/43

If y(x) is a solution of

$x2y" - 4xy' + 6y = 0, \ \ \ \ y(-1) = 1, \ \ \ \ y'(-1) = 0$

Then the value of y(2) is __28___ .

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By Cauchy Eulers homogeneous linear differential equations-

$x2y" - 4xy' + 6y = 0$

D(D-1)y-4Dy+6y=0

Where , $x=e^z$

D=\frac{d}{dz}

$(D^2-5D+6)y=0$ Assume, D=m

$(m^2-5m+6)y=0$

$(m^2-2m-3m+6)y=0$

(m-2)(m-3)=0

Now, Solution $y=c_1 e^2z+c_2 e^3z$

$y=c_1 x^2+c_2 x^3$

y(-1)=1

$1=c_1-c_2$ ,

$c_1=1+c_2$

${y}'(-1)=0$

${y}'=2c_1 x+3 c_2 x^2$

$0=-2 c_1+3 c_2$

Slove both, c1=3 and c2=2

Then , $y(2)=4 c_1+8 c_2$ y(2)=28 (Ans)