Variation of water depth ( {y} ) in a gradually varied open channel flow is given by the first order differential equation \frac{d y}{d x}=\binom{1-e^{-\frac{10}{3} In(y)}}/{250-45 e^{-3 In(y)}} Given initial conditions: y(x=0)=0.8 {m}. The depth (in {m}, up to three decimal places) of flow at a downstream section at x=1 from one calculation step of Single Step Euler Method is

GATE Civil Engineering (CE) 2018 Shift 1 Solved Paper

Question : 37 of 65

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Variation of water depth (

{y}

) in a gradually varied open channel flow is given by the first order differential equation

\;{d y}/{d x}=\;{1-e^{-{10}/{3} In(y)}}/{250-45 e^{-3 In(y)}}

Given initial conditions:

y(x=0)=0.8 {m}

.
The depth (in

{m}

, up to three decimal places) of flow at a downstream section at

x=1

from one calculation step of Single Step Euler Method is

Solution:

\;{d y}/{d x}=\;{1-e^{-{10}/{3} \ln y}}/{250-45 e^{-3 In y}}

y_{1}=y_{0}+hf(x_{0}, y_{0})

⇒ \;\; y_{1}=0.8+1[\;{1-e^{-{10}/{3} \ln 0.8}}/{250-45 e^{-3 \ln 0.8}}]

=0.8+1(\;{-1.1039}/{162.109})

=0.793 {m}

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