Solve the ODE

Use the Euler forward method $($forward differencing$)$

Consider the following values of

B $=0$

B $=-0\mathrm{.}199$

B $=0\mathrm{.}8$

Discretization

Solve it on the interval:

$0<=$n $<=10$with n$\_$n $=1000$points

discretize eta

define start value of eta: eta$\_$s

define end value of eta: eta$\_$e

define number of points: n$\_$eta

discretize eta

preallocate arrays to store solution

store solution for beta $=0$case in yb$1$

store solution for beta $=-0\mathrm{.}199$case in yb$2$

store solution for beta $=0\mathrm{.}8$case in yb$3$

NOTE: each solution vector needs to hold n$\_$eta rows and $3$columns

where n$\_$eta is the number of points used to discretize eta

define ode function to update solution

define an anonymous function $($function handle$)$with thefollowing properties

name: dydeta

inputs: y and beta, where y is a row vector with $3$values and beta is a scalar

ouptuts: a row vector with the rate of change of the state defined with the reduced order system above

ode check

define a row vector names y$\_$check with the values: $1,2$and $3$

assign a variable named beta$\_$check the value $-1$

call the previously define function dydeta with the inputs y$\_$check and beta$\_$check and store the result in dy$\_$check

solve the ode for case $1($beta $=0)$

assign beta value

apply initial condition provided for beta $=0$

solve the ode using forward differencing

solve the ode for case $2($beta $=-0\mathrm{.}199)$

assign beta value

set initial condition for beta $=-0\mathrm{.}199$

solve the ode using forward differencing

solve ode for case $3($beta $=0\mathrm{.}8)$

assign beta value

set initial condition for beta $=0\mathrm{.}8$

solve the ode using forward differencing

plot results

plot the second column of yb$1$over eta as a blue line with line width $4$

in the same plot add:

the second column of yb$2$over eta as a red line with line width $4$

the second column of yb$2$over eta as a green line with line width $4$

customize the plot

$-$switch on the grid

$-$set font size to $20$

Reduced order system:

y$`1=$y$2$

y$`2=$y$3$

y$`3=-$y$1($y$3)-$B$[1-($y$2)^2]$

Initial conditions:

B $=0$case:

y$1($n$=0)=0$

y$2($n$=0)=0$

y$3($n$=0)=0\mathrm{.}4696$

B $=-0\mathrm{.}199$case:

y$1($n$=0)=0$

y$2($n$=0)=0$

y$3($n$=0)=-0\mathrm{.}0008$

B $=0\mathrm{.}8$case:

y$1($n$=0)=0$

y$2($n$=0)=0$

y$3($n$=0)=1\mathrm{.}1172$

MATLAB code