Solving simple integro-differential equation using dsolve in Matlab

I have following integro-differential equation:

I tried to solve it using symbolic math toolbox in Matlab:

syms x(t)
Dx = diff(x,t)
cond = x(0)==1
eqn = Dx + 2*x + 2*int(x) == 0
x(t) = dsolve(eqn, cond)

dsolve gives:

x(t) =
 
exp(1 - (2*t + 2)^2/4)

Matlab solution is wrong.

Solving equation by hand using Laplace transform I get:

Why Matlab gives the wrong symbolic solution?

Thank you.

Answer 1

[Torsten]

If you insert the line

simplify(diff(x,t)+2*x+2*x*t)

you will see that this expression turns out to be equal to 0. So MATLAB interprets x(t) as constant over [0 t] in the integration.

I'd say that "dsolve" is not suited to solve integro-differential equations or - since you don't get an error message - it's a bug.

[Paul]

Not a solution to the problem, but worth pointing out that the code does not properly reflect the actual equation in the question.

syms x(t)
Dx = diff(x,t);
cond = x(0)==1;
eqn = Dx + 2*x + 2*int(x) == 0

eqn(t) = 

Here, int(x) is an anti-derivative of x(t), which is not the same as the integral in the problem statement.

Consequently, further operations on eqn may not yield the expected result. For example, taking the Laplace transform does not apply the expected "integral rule" to the third term.

laplace(eqn)

ans = 

The equation should be entered as

syms tau
eqn = Dx + 2*x + 2*int(x(tau),tau,0,t) == 0

eqn(t) = 

The output from dsolve still not neccessarily helpful

sol(t) = dsolve(eqn,cond)

sol(t) = 

But now laplace() works and we can solve the problem that way

Leqn = laplace(eqn)

Leqn = 

syms s
L(s) = rhs(isolate(Leqn,laplace(x,t,s)))

L(s) = 

x(t) = ilaplace(subs(L(s),x(0),1))

x(t) =