Answer to Question #278484 in Quantitative Methods for Tumi

Question

.1.1 Use Euler’s method with step size h = 0.1 to approximate the solution to the initial valueproblem: y'=x y, y(1)=4, atthepointsx=1.1,1.2and1.3,correcttofivedecimal

places.

2.1.2 If the analytical solution to the initial value problem in (2.1.1) is

(4) y = 1 (x2 + 7), determine

4

the %Error in the numerical method where x =1.3

2.2 A chicken cools down from 100 °C to 60 °C within 10 

Answer 1

2.1.1

$y'=x\sqrt y,y(1)=4$

Euler method

$y_1=y_0+hf(x_0,y_0)=4+0.1f(1,4)=4.2$

 $y_2=y_1+hf(x_1,y_1)=4.2+0.1f(1.1,4.2)=4.42543$

$y_3=y_2+hf(x_2,y_2)=4.42543+0.1f(1.2,4.42543)=4.67787$

$y(1.3)=4.67787$

2.1.2

$\sqrt y=(x^2+7)/4$

$y(1.3)=(1.3^2+7)^2/16=4.71976$

error = $\frac{4.71976-4.67787}{4.71976}=0.0089=0.89\%$

2.2

A chicken cools down from 100 °C to 60 °C within 10 minutes of being placed in a stream of air

at $20\degree C$

by Newton's Law of Cooling:

rate of cooling:

$\frac{dT}{dt}=-k(T-T_{env})$

where k is constant,

T_env is temperature of the environment

then:

$\frac{dT}{dt}=-k(T-20)$

$T=ce^{-kt}-20$

$T(0)=c-20=100$

$c=120$

$T(10)=120e^{-10k}-20=60$

$k=-ln(80/120)/10=0.04$

$T(t)=120e^{-0.04t}-20$