MATLAB: How to use initial conditions with an equation

equationgraphinitial conditionsodeplotting

Hello I'm wondering how can I use the given initial conditions to an equation e.g. x=0 and y=1 to be used on an equation e.g. Y=2*x^3+6*x^2-10*x+4 and its derivative dydx=6*x^2+12x-10 when I try to plot Y and dydx from 0 to 6 with 0.1 steps. Do I need a for loop or something of that matter?

Best Answer

The easiest way is to use polyval:

% Y=2*x^3+6*x^2-10*x+4
% dydx=6*x^2+12x-10
Y = [2  6  -10  4];         % EXPRESS COEFFICIENTS AS A VECTOR
dydx = [6  12  -10];
x = 0:0.1:6;
Y_x = polyval(Y,x);
dydx_x = polyval(dydx,x);
figure(1)
plot(x, Y_x,  x,dydx_x)
grid
legend('Y', '^{dy}/_{dx}', 'Location', 'NW')

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You can transform your equation d2y/dx2=.-5dy/dx-7y to get:

y1=y
y2=dy/dx
dy1=dy/dx=y2
dy2=d2y/d2x=-5dy/dx-7y=-5y2-7y1

Create a file named myode.m

function dy=myode(x,y)
dy=[0;0]
dy(1)=y(2)
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Then run these lines

yin=[0 4 ]  % initial values
time=[0 5]
[x,y]=ode45(@myode,time,yin)
plot(x,y)

MATLAB: Solve a system of second order coupled differential equations with bvp4c

In the definition of bvp_function, when MATLAB reaches the line

dYdx(2) = (Y(2) + Y(4)) * (num1 - Y(2) - dYdx(3)/Y(3) - dYdx(4));

at that time, the values of dYdx(3) and dYdx(4) are undefined, so MATLAB throws an error.

You can solve this problem, either by changing the order of calculation like this

function dYdx = bvp_function(x, Y)
% Y(1)=j, Y(2)=j', Y(3) = xsi, Y(4) = t 

    num1 = 2.331;
    dYdx(4) = -Y(2)*exp(Y(1));
    dYdx(3) = Y(2) + dYdx(4) - Y(3);
    dYdx(1) = Y(2);
    dYdx(2) = (Y(2) + Y(4)) * (num1 - Y(2) - dYdx(3)/Y(3) + dYdx(4));
    dYdx = dYdx(:);
end

or replace the values of c' and t' in equation 1 and 2 like this

function dYdx = bvp_function(x, Y)
% Y(1)=j, Y(2)=j', Y(3) = xsi, Y(4) = t 
    num1 = 2.331;
    dYdx(1) = Y(2);
    dYdx(2) = (Y(2) + Y(4)) * (num1 - Y(2) - (Y(2) - Y(2)*exp(Y(1)) - Y(3))/Y(3) + Y(2)*exp(Y(1)));             
    dYdx(3) = Y(2) - Y(2)*exp(Y(1)) - Y(3);
    dYdx(4) = -Y(2)*exp(Y(1));
    dYdx = dYdx(:);
end

However, even after this, there seems to be a singularity in your system of ODEs. You need to analyze your equations to see what might be causing the singularity.

Best Answer

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MATLAB: Solve a second order differential equation in matlab

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