MATLAB: Solving ODE, 2nd degree using dsolve function. How to simplify the answer

dsolvesimplify

The answer I am getting is exponential with an iota as the power. I want to get in terms if sin and cos and a simplified answer. How do I do it?

Best Answer

I am not certain exactly what you want.

To simplify a result, use the simplify 'steps' option, and to express a result in terms of sine and cosine, use rewrite with the 'sincos' option:

syms A w f1(t) f2(t)
f1(t) = 1;
F1(w) = int(f1*exp(1i*w*t), -1, 1)
F1(w) = simplify(F1(w), 'steps', 10)
f2 = t;
F2(w) = int(f2*exp(1i*w*t), 0, 1)
F2(w) = rewrite(F2(w), 'sincos')

producing:

F1(w) =
(2^(1/2)*sin(w)*1i)/((w*1i)^(1/2)*((w*1i)/2)^(1/2))
F1(w) =
(2*sin(w))/w

and:

F2(w) =
- exp(t*1i)*1i + 1i
F2(w) =
sin(t) - cos(t)*1i + 1i

Related Solutions

MATLAB: Surface Plot for ODE solution

You need to make a few changes in your code:

w = 1; 
k=1;
figure
tspan = linspace(0, 5);                                 % Create Constant ‘tspan’
zv=0.1:0.01:0.5;                                        % Vector Of ‘z’ Values
gs2 = zeros(numel(tspan), numel(zv));                   % Preallocate
for k = 1:numel(zv)
    z = zv(k);
f = @(t,x) [-1i.*(2*w + 2*z).*x(1) + -1i.*sqrt(2).*k.*x(2);-1i.*sqrt(2).*k.*x(1) + -1i.*2*w*x(2)+-1i.*sqrt(2).*k.*x(3);-1i.*sqrt(2).*k.*x(2)+-1i.*2*w*x(3)];
[t,xa] = ode45(f,tspan,[0 1 0]);
gs = abs(xa).^2;
gs2(:,k) = gs(:,2);                                     % Save Second Column Of ‘gs’ In ‘gs2’ Matrix
end
figure
surf(t,zv,gs2')
grid on
xlabel('t')
ylabel('z')
shading('interp')

Experiment to get the result you want.

MATLAB: Cant Solve ODE with dsolve

So you have a linear first-order differential equation looking something like this:

Then I'd suggest that special cases are combinations of A1 B1 and w equals zero. If I run your code without the numerical values of A1 B1 phi and w I get:

TSol = dsolve(diff(T) == (A1*exp(1i*(w*t+phi))+A1)*(B1*exp(1i*w*t)-T),cond)
TSol =
10*exp(-(A1*exp(phi*1i)*1i)/w)*exp(- A1*t + (A1*exp(phi*1i)*exp(t*w*1i)*1i)/w) + exp(- A1*t + (A1*exp(phi*1i)*exp(t*w*1i)*1i)/w)*int(A1*B1*exp(A1*u + u*w*1i - (A1*exp(phi*1i)*exp(u*w*1i)*1i)/w)*(exp(phi*1i + u*w*1i) + 1), u, 0, t, 'IgnoreSpecialCases', true, 'IgnoreAnalyticConstraints', true)
>> pretty(TSol)
                               t
                               /
   /   A1 #3 1i \             |           /             A1 #3 exp(#1) 1i \
exp| - -------- | #2 10 + #2  |  A1 B1 exp| A1 u + #1 - ---------------- | (exp(phi 1i + #1) + 1) du
   \       w    /            /            \                     w        /
                               0
where
   #1 == u w 1i
            /          A1 #3 exp(t w 1i) 1i \
   #2 == exp| - A1 t + -------------------- |
            \                    w          /
   #3 == exp(phi 1i)

It is a long expression, but it containe nothing extraordinarily complicated, but if for example w is zero this falls appart. You should be able to convert this into a regular m-function in a couple of minutes. It seems to fit the ode rather nicely.

HTH

Best Answer

Related Solutions

MATLAB: Surface Plot for ODE solution

MATLAB: Cant Solve ODE with dsolve

Related Question