Solve a second order differential equation

hi.i want to solve this second order differential equation in matlab an plot figure. can you help me please?

d2a/de2=(((((2+c2)./(1+c2)).*(Fu.^2))+1)*(a.^c2)-((2+c2)./(1+c2)).*(Fu.^2./a)-a.^(2+(2.*c2)))./( ((2+c2)/((3+c2).*(1+c2))).*Fu.^2 );

Fu=1

c2=0,0.5,1

Answer 1

[Ameer Hamza]

This is a second-order ODE which can be solved using ode45(). See the example titled "Solve Nonstiff Equation" here: https://www.mathworks.com/help/matlab/ref/ode45.html#bu3uj8b

[Mj]

thanks alot. i am beginner in matlab.is it possible for you to send me the code?

[Ameer Hamza]

Can you write your equation in MATLAB form? Then It will be easier for me to write a solution.

[Mj]

yes. the equation is :

x=(((((2+c2)./(1+c2)).*(Fu.^2))+1)*(a.^c2)-((2+c2)./(1+c2)).*(Fu.^2./a)-a.^(2+(2.*c2)))./( ((2+c2)/((3+c2).*(1+c2))).*Fu.^2 )

x=d2a/de2

Fu=1

c2=0,0.5,1

[Ameer Hamza]

What are the initial conditions? Following code shows an example

Fu=1;
c2=0;

dAdt = @(t, A) [A(2);
    (((2+c2)./(1+c2).*Fu.^2+1)*A(1).^c2-(2+c2)./(1+c2).*Fu.^2./A(1)-A(1).^(2+2.*c2)) ./( (2+c2)/((3+c2).*(1+c2)).*Fu.^2 )];

tspan = [-2 0];
IC = [1; -0.1];
[t, A] = ode45(dAdt, tspan, IC);

plot(t, A(:,1), '-o');
legend({'$\alpha$'}, 'Interpreter', 'latex', ...
    'FontSize', 16, 'Location', 'best')

[Mj]

does it need initial condition? what initial condition did you use in this code?

[Ameer Hamza]

This is the line for initial condition

IC = [1; -0.1];

First is the value for alpha and second is for d_alpha. Initial condition is applied at left side of interval in tspan.

[Mj]

is it possible to solve this by fourth order Rung-kutta method?

[Ameer Hamza]

ode45() also use RK-(4,5) algorithm as desribed here: https://www.mathworks.com/help/matlab/ref/ode45.html#bu0200e-1

[Mj]

i put my initial condition instead of yours, but it did not give me the righ figure.

IC = [0.75; 0];

The alpha value as the initial condition is 0.75. But I do not have d-Alpha in the same place. Elsewhere, d-alpha value is zero.

do you know what should i do?

excuse me because of my boring questions.

[Ameer Hamza]

This is the solution according to the equation you shared. Check if there is a mistake in writing the equation.