How to solve nonlinear equation?

Hello,

I wrote the following code to derive an analytical solution to nonlinear equation but it gives an error. Could you please help me to fix it? Or any suggestion to solve in an analytical way. Thanks

syms x(t);
ode = diff(x,t) == -1*(1-abs(x)^2*x-(1-0.5)*x);
cond = x(0) == 1;
xSol(t) = dsolve(ode,cond);

Warning: Unable to find symbolic solution.

t = 0:1:100;
xSols = xSol(t);

plot(t,xSols)

Error using plot
Invalid data argument.

[Torsten]

If it helps: You can get t as an analytical function of x, but I think it's not possible to solve for x.

Answer 1

[Sam Chak]

Hi @Semiha

I'm afraid that the nonlinear differential equation may not have an analytical solution. In such cases, you can utilize the 'ode45' solver to obtain a numerical solution.

ode     = @(t, x) 1*(1 - abs(x)^2*x - (1 - 0.5)*x);
tspan   = [0 10];       % simulation time
x0      = 1;            % initial value
options = odeset('RelTol', 1e-4, 'AbsTol', 1e-6);
[t, x]  = ode45(ode, tspan, x0, options);

plot(t, x), grid on, xlabel('t'), ylabel('x(t)')

[Sam Chak]

Or, you can try finding an implicit solution:

syms x(t);
ode     = diff(x,t) == 1 - x^3 - 0.5*x;
cond    =      x(0) == 1;
xSol(t) = dsolve(ode, cond, 'Implicit', true)

xSol(t) = 

[Semiha]

I have already derive a numerical solution by using ode45 in matlab and also in mathematica. I need to derive an analytical solution. Yes, maybe there is no an explict analytical solution. But at least I tried to find plot(lxl,t), if it is possible.

Thank you for your kind response.

[Torsten]

syms x(t) u
ode     = diff(x,t) == 1 - x^3 - 0.5*x;
cond    =      x(0) == 1;
xSol = dsolve(ode, cond, 'Implicit', true);
xSol = subs(xSol,x,u);
T = 0:0.1:10;
X = arrayfun(@(T)vpasolve(subs(xSol,t,T),u),T)

X = 

plot(T,X)

Warning: Imaginary parts of complex X and/or Y arguments ignored.

grid on

[Semiha]

Thank you so much for your response.

A question come to my mind, what If I turn to equation diff(x,t) == i(1 - x^3 - 0.5*x); where i is imaginary it gives error.

This solution is valid only for the real functions?

[Semiha]

I mean diff(x,t) == i(1 - x^3 - 0.5*x) and x(0)=0

[Torsten]

I don't know why for the symbolic solution, not for all t-values solutions for x are returned.

ode     = @(t, x) 1i*(1 - x^3 - 0.5*x);
tspan   = [0 10];       % simulation time
x0      = 0;            % initial value
[t, x]  = ode45(ode, tspan, x0);
figure(1)
plot(t, real(x)), grid on, xlabel('t'), ylabel('real(x(t))')

figure(2)
plot(t, imag(x)), grid on, xlabel('t'), ylabel('imag(x(t))')

syms x(t) u
ode     = diff(x,t) == 1i*(1 - x^3 - 0.5*x);
cond    =      x(0) == 0;
xSol = dsolve(ode, cond, 'Implicit', true);
xSol = subs(xSol,x,u);
vpasolve(subs(xSol,t,1),u)

ans =