MATLAB: Symbolic ODE solver Error

errorodesymbolic

Wrong answer!

>> simplify( dsolve('Dy + y * cos(x) = sin(x) * cos(x)', 'y(0) = 0') )
ans =
                             sin(2 x) (-1 + exp(-cos(x) t))
                       - 1/2 ------------------------------
                                         cos(x)
  >>

The argument of the exponential should be sin; also, where's the t coming from?

Best Answer

Try

syms y(x)
eqn(x) = diff(y(x), x)+y(x)*cos(x) == sin(x)*cos(x);
dsolve(eqn,y(0)==0)

Note: the above includes a work-around for a bug in current versions of the symbolic toolbox. If you were to assign to eqn instead of to eqn(x) then the dsolve() would fail.

I am not sure why it produces the result it does for the literal case; it is probably a bug.

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MATLAB: Using dsolve function to solve this equation

For grins, I decided to see what a Laplace transform solution would produce:

syms s y(t) y0 Y
Lsin = laplace(sin(t));
Eqn1 = Y*(s-1) == Lsin;
Eqn2 = solve(Eqn1, Y);
Eqn3 = ilaplace(partfrac(Eqn2))
Eqn4 = simplify(Eqn3, 'steps',10)
Eqn3 =
exp(t)/2 - cos(t)/2 - sin(t)/2
Eqn4 =
exp(t)/2 - (2^(1/2)*sin(t + pi/4))/2

MATLAB: Problem with simplify function used with matrices

MATLAB's symbolic toolbox is not perfect (perhaps it is impossible to make a perfect symbolic engine), and every year they make some improvements. This seems one such case. Also, "simplification" is not a well defined idea. The tips section of simplify() documentation mentions: "Simplification of mathematical expression is not a clearly defined subject. There is no universal idea as to which form of an expression is simplest. The form of a mathematical expression that is simplest for one problem might be complicated or even unsuitable for another problem."

Are you using an older release of MATLAB? In R2020b, it is simplified as you described in your comment

syms t1 t2 t3 t4 a1 a2 d3
M = [
cos(t4)*(cos(t1)*cos(t2) - sin(t1)*sin(t2)) - sin(t4)*(cos(t1)*sin(t2) + cos(t2)*sin(t1)), - cos(t4)*(cos(t1)*sin(t2) + cos(t2)*sin(t1)) - sin(t4)*(cos(t1)*cos(t2) - sin(t1)*sin(t2)), 0, a1*cos(t1) + a2*cos(t1)*cos(t2) - a2*sin(t1)*sin(t2)
cos(t4)*(cos(t1)*sin(t2) + cos(t2)*sin(t1)) + sin(t4)*(cos(t1)*cos(t2) - sin(t1)*sin(t2)),   cos(t4)*(cos(t1)*cos(t2) - sin(t1)*sin(t2)) - sin(t4)*(cos(t1)*sin(t2) + cos(t2)*sin(t1)), 0, a1*sin(t1) + a2*cos(t1)*sin(t2) + a2*cos(t2)*sin(t1)
                                                                                        0,                                                                                           0, 1,                                                   d3
                                                                                        0,                                                                                           0, 0,                                                   1];
                                                                                    
M_simplify = simplify(M)

Result

>> M_simplify
M_simplify =
[cos(t1 + t2 + t4), -sin(t1 + t2 + t4), 0, a2*cos(t1 + t2) + a1*cos(t1)]
[sin(t1 + t2 + t4),  cos(t1 + t2 + t4), 0, a2*sin(t1 + t2) + a1*sin(t1)]
[                0,                  0, 1,                           d3]
[                0,                  0, 0,                            1]

You may experiment with following option if you are using an older release of MATLAB

M_simplify = simplify(M, 'Seconds', 10); % specify maximum number of seconds MATLAB spent on simplification

Best Answer

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MATLAB: Using dsolve function to solve this equation

MATLAB: Problem with simplify function used with matrices

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