A differential equation is given as x^{2} {d^{2} y}/{d x^{2}}-2 x \frac{d y}{d x}+2 y=4 The solution of the differential equation in terms of arbitrary constants C_{1} and C_{2} is

Correct Answer is : y=C1x2+C2x+2

GATE Mechanical Engineering (ME) 2019 Shift 2 Solved Paper

Question : 60 of 65

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A differential equation is given as

x^{2} \;{d^{2} y}/{d x^{2}}-2 x \;{d y}/{d x}+2 y=4

The solution of the differential equation in terms of arbitrary constants

C_{1}

and

C_{2}

Solution:

Given differential equation

(D(D-1)-2 D+2) y=4

where

x=e^{z}

Auxiliary equation,

(D-2)(D-1)=0

D=1,2

C F=C_{1} e^{1 x}+C_{2} e^{2 x}

P I=\;{4}/{(D-2)(D-1)}=2

Y=C F+P I

∴ y=C_{1} e^{1 x}+C_{2} e^{2 x}+2

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