If y is the solution of the differential equation y^{3} \frac{d y}{d x}+x^{3}=0, y(0)=1, the value of y(-1) is

GATE Mechanical Engineering (ME) 2018 Shift 2 Solved Paper

Question : 24 of 65

Marks: +1, -0

y

is the solution of the differential equation

y^{3} \;{d y}/{d x}+x^{3}=0, y(0)=1

, the value of

y(-1)

Solution:

y^{3} \;{d y}/{d x}=-x^{3}

y^{3} d y \;\; =-x^{3} d x

∫ y^{3} d y \;\; =-∫ x^{3} d x

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

\;{x^{4}+y^{4}}/{4}=C

y(0)=1

\;{0+1}/{4}=C

C=\;{1}/{4}

x^{4}+y^{4} \;\; =1

y^{4} \;\; =1-x^{4}

y \;\; =√^{4}{1-x^{4}}

y(-1) \;\; =0

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