Consider the homogeneous ordinary differential equation x^{2} {d^{2} y}/{d x^{2}}-3 x \frac{d y}{d x}+3 y=0, x>0 with {y}( {x}) as a general solution. Given that y(1)=1 and y(2)=14 the value of y(1.5) rounded off to two decimal places, is ________________.

GATE Electronics and Communications (EC) 2019 Solved Paper

Question : 59 of 65

Marks: +1, -0

Consider the homogeneous ordinary differential equation

x^{2} \;{d^{2} y}/{d x^{2}}-3 x \;{d y}/{d x}+3 y=0, x>0

with

{y}( {x})

as a general solution. Given that

y(1)=1

and

y(2)=14

the value of

y(1.5)

rounded off to two decimal places, is ________________.

Solution:

. Given differential equation is of Cauchy - Euler differential equation type.
So let

{x}= {e}^{2}

∴ z=\ln x

The differential equation can be written as,

{D}( {D}-1)-3 {D}+3=0

∴ {D}^{2}-4 {D}+3=0

∴ {D}=1,3

∴ {y}= {C}_{1} {e}^{2}+ {C}_{2} {e}^{3 {z}}

∴ {y}= {C}_{1} {x}+ {C}_{2} {x}^{3}

Now

y(1)=1

∴ {C}_{1}+ {C}_{2}=1

.....(i)
And

{y}(2)=14

∴ 2 C_{1}+8 C_{2}=14

.....(ii)
From (i) and (ii)

{C}_{1}=-1, {C}_{2}=2

∴ y=-x+2 x^{3}

∴ y(1.5)=-1.5+2(1.5)^{3}

∴ {y}(1.5)=5.25

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